INTERNATIONAL GONGRESS >L is an invertible sheaf on A A AAA, then we have the Weil height h A , L h A , L h_(A,L)h_{A, \mathscr{L}}hA,L from Section 2.1. Recall that h A , L h A , L h_(A,L)h_{A, \mathscr{L}}hA,L is only defined up to addition of a bounded function on A ( Q ¯ ) A ( Q ¯ ) A( bar(Q))A(\overline{\mathbb{Q}})A(Q¯). For abelian varieties, there is a canonical choice of function in the equivalence class h A , L h A , L h_(A,L)h_{A, \mathscr{L}}hA,L called the canonical or Néron-Tate height. A general reference for this section is [9, CHAPTER 9].
For an integer n Z n ∈ Z n inZn \in \mathbb{Z}n∈Z, let [ n ] [ n ] [n][n][n] denote the multiplication-by- n n nnn endomorphism of A A AAA. Then L L L\mathscr{L}L is called even or symmetric if there is an isomorphism [ 1 ] L L [ − 1 ] ∗ L ≅ L [-1]^(**)L~=L[-1]^{*} \mathscr{L} \cong \mathscr{L}[−1]∗L≅L. It is called odd or antisymmetric if [ 1 ] L L ( 1 ) [ − 1 ] ∗ L ≅ L ⊗ ( − 1 ) [-1]^(**)L~=Lox(-1)[-1]^{*} \mathscr{L} \cong \mathscr{L} \otimes(-1)[−1]∗L≅L⊗(−1). If L L L\mathscr{L}L is any ample invertible sheaf on A A AAA, then L [ 1 ] L L ⊗ [ − 1 ] ∗ L Lox[-1]^(**)L\mathscr{L} \otimes[-1]^{*} \mathscr{L}L⊗[−1]∗L is ample and even. So any abelian variety admits an even, ample invertible sheaf.
Suppose that L L L\mathscr{L}L is even. Then [2]* L L 4 L ≅ L ⊗ 4 L~=L^(ox4)\mathscr{L} \cong \mathscr{L}^{\otimes 4}L≅L⊗4 is a consequence of the Theorem of the Cube. So Theorem 2.4 implies h A , L [ 2 ] = 4 h A , L h A , L ∘ [ 2 ] = 4 h A , L h_(A,L)@[2]=4h_(A,L)h_{A, \mathscr{L}} \circ[2]=4 h_{A, \mathscr{L}}hA,L∘[2]=4hA,L as classes and by iteration h A , L [ 2 k ] = h A , L ∘ 2 k = h_(A,L)@[2^(k)]=h_{A, \mathscr{L}} \circ\left[2^{k}\right]=hA,L∘[2k]= 4 k h A , L 4 k h A , L 4^(k)h_(A,L)4^{k} h_{A, \mathscr{L}}4khA,L for all k 1 k ≥ 1 k >= 1k \geq 1k≥1. We fix a representative h A , L h A , L ′ h_(A,L)^(')h_{A, \mathscr{L}}^{\prime}hA,L′ of h A , L h A , L h_(A,L)h_{A, \mathscr{L}}hA,L and find h A , L [ 2 k ] = h A , L ′ ∘ 2 k = h_(A,L)^(')@[2^(k)]=h_{A, \mathscr{L}}^{\prime} \circ\left[2^{k}\right]=hA,L′∘[2k]= 4 k h A , L + O k ( 1 ) 4 k h A , L ′ + O k ( 1 ) 4^(k)h_(A,L)^(')+O_(k)(1)4^{k} h_{A, \mathscr{L}}^{\prime}+O_{k}(1)4khA,L′+Ok(1) on A ( Q ¯ ) A ( Q ¯ ) A( bar(Q))A(\overline{\mathbb{Q}})A(Q¯). Tate's Limit Argument is used to show convergence in the following definition.
Definition 2.6. Let L L L\mathscr{L}L be an even invertible sheaf on A A AAA and let P A ( Q ¯ ) P ∈ A ( Q ¯ ) P in A( bar(Q))P \in A(\overline{\mathbb{Q}})P∈A(Q¯). Then the limit
(2.3) h ^ A , L ( P ) = lim k h A , L ( P ) 4 k (2.3) h ^ A , L ( P ) = lim k → ∞   h A , L ′ ( P ) 4 k {:(2.3) hat(h)_(A,L)(P)=lim_(k rarr oo)(h_(A,L)^(')(P))/(4^(k)):}\begin{equation*} \hat{h}_{A, \mathscr{L}}(P)=\lim _{k \rightarrow \infty} \frac{h_{A, \mathscr{L}}^{\prime}(P)}{4^{k}} \tag{2.3} \end{equation*}(2.3)h^A,L(P)=limk→∞hA,L′(P)4k
exists and is independent of the choice of representative h A , L h A , L ′ h_(A,L)^(')h_{A, \mathscr{L}}^{\prime}hA,L′ of h A , L h A , L h_(A,L)h_{A, \mathscr{L}}hA,L. The real-valued function P h ^ A , L ( P ) P ↦ h ^ A , L ( P ) P|-> hat(h)_(A,L)(P)P \mapsto \hat{h}_{A, \mathscr{L}}(P)P↦h^A,L(P) is called the canonical or Néron-Tate height (on A A AAA attached to L L L\mathscr{L}L ).
If L L L\mathscr{L}L is even, then (2.3) immediately implies h ^ A , L ( [ 2 ] ( P ) ) = 4 h ^ A , L ( P ) h ^ A , L ( [ 2 ] ( P ) ) = 4 h ^ A , L ( P ) hat(h)_(A,L)([2](P))=4 hat(h)_(A,L)(P)\hat{h}_{A, \mathscr{L}}([2](P))=4 \hat{h}_{A, \mathscr{L}}(P)h^A,L([2](P))=4h^A,L(P) for all P A ( Q ¯ ) P ∈ A ( Q ¯ ) P in A( bar(Q))P \in A(\overline{\mathbb{Q}})P∈A(Q¯). If P P PPP has finite order, then [ 2 m ] ( P ) = [ 2 n ] ( P ) 2 m ( P ) = 2 n ( P ) [2^(m)](P)=[2^(n)](P)\left[2^{m}\right](P)=\left[2^{n}\right](P)[2m](P)=[2n](P) for distinct integers 0 m < n 0 ≤ m < n 0 <= m < n0 \leq m<n0≤m<n by the Pigeonhole Principle. Thus h ^ A , L ( P ) = 0 h ^ A , L ( P ) = 0 hat(h)_(A,L)(P)=0\hat{h}_{A, \mathscr{L}}(P)=0h^A,L(P)=0.
There is nothing special about [2]. Indeed, one can replace [2] by [ m [ m [m[\mathrm{m}[m ] in (2.3) for any integer m 2 m ≥ 2 m >= 2m \geq 2m≥2; one then needs to replace 4 k 4 k 4^(k)4^{k}4k in the denominator by m 2 k m 2 k m^(2k)m^{2 k}m2k.
What happens if L L L\mathscr{L}L is an odd invertible sheaf? In this case, [2] L L 2 ∗ L ≅ L ⊗ 2 ^(**)L~=L^(ox2){ }^{*} \mathscr{L} \cong \mathscr{L}^{\otimes 2}∗L≅L⊗2. Then a similar limit (2.3) exists, but now we need to divide by 2 k 2 k 2^(k)2^{k}2k.
The set of odd invertible sheaves is a divisible subgroup of Pic ( A ) Pic ⁡ ( A ) Pic(A)\operatorname{Pic}(A)Pic⁡(A). From this, one can show that, after possibly extending the base field F F FFF, any invertible sheaf L L L\mathscr{L}L on A A AAA decom-
poses as L + L L + ⊗ L − L_(+)oxL_(-)\mathscr{L}_{+} \otimes \mathscr{L}_{-}L+⊗L−with L + L + L_(+)\mathscr{L}_{+}L+even and L L L\mathscr{L}L - odd. One then defines h ^ A , L = h ^ A , L + + h ^ A , L h ^ A , L = h ^ A , L + + h ^ A , L hat(h)_(A,L)= hat(h)_(A,L_(+))+ hat(h)_(A,L)\hat{h}_{A, \mathscr{L}}=\hat{h}_{A, \mathscr{L}_{+}}+\hat{h}_{A, \mathscr{L}}h^A,L=h^A,L++h^A,L; the decomposition of L L L\mathscr{L}L is not quite unique, but this ambiguity does not affect h ^ A , L h ^ A , L hat(h)_(A,L)\hat{h}_{A, \mathscr{L}}h^A,L.
For our purposes, we often restrict to even invertible sheaves.
Let us collect the some important facts about the Néron-Tate height.
Theorem 2.7. Let us keep the notation above. In particular, A A AAA is an abelian variety defined over a number field F Q ¯ F ⊆ Q ¯ F sube bar(Q)F \subseteq \overline{\mathbb{Q}}F⊆Q¯.
(i) Then association L h ^ A , L L ↦ h ^ A , L L|-> hat(h)_(A,L)\mathscr{L} \mapsto \hat{h}_{A, \mathscr{L}}L↦h^A,L is a group homomorphism from Pic ( V ) Pic ⁡ ( V ) Pic(V)\operatorname{Pic}(V)Pic⁡(V) to the additive group of real-valued maps A ( Q ¯ ) R A ( Q ¯ ) → R A( bar(Q))rarrRA(\overline{\mathbb{Q}}) \rightarrow \mathbb{R}A(Q¯)→R.
Suppose L L L\mathscr{L}L is an invertible sheaf on A A AAA.
(ii) The Néron-Tate height h ^ A , L h ^ A , L hat(h)_(A,L)\hat{h}_{A, \mathscr{L}}h^A,L represents the Weil height h A , L h A , L h_(A,L)h_{A, \mathscr{L}}hA,L.
(iii) If L L L\mathscr{L}L is even, then the parallelogram equality
h ^ A , L ( P + Q ) + h ^ A , L ( P Q ) = 2 h ^ A , L ( P ) + 2 h ^ A , L ( Q ) h ^ A , L ( P + Q ) + h ^ A , L ( P − Q ) = 2 h ^ A , L ( P ) + 2 h ^ A , L ( Q ) hat(h)_(A,L)(P+Q)+ hat(h)_(A,L)(P-Q)=2 hat(h)_(A,L)(P)+2 hat(h)_(A,L)(Q)\hat{h}_{A, \mathscr{L}}(P+Q)+\hat{h}_{A, \mathscr{L}}(P-Q)=2 \hat{h}_{A, \mathscr{L}}(P)+2 \hat{h}_{A, \mathscr{L}}(Q)h^A,L(P+Q)+h^A,L(P−Q)=2h^A,L(P)+2h^A,L(Q)
holds for all P , Q A ( Q ¯ ) P , Q ∈ A ( Q ¯ ) P,Q in A( bar(Q))P, Q \in A(\overline{\mathbb{Q}})P,Q∈A(Q¯).
(iv) If L L L\mathscr{L}L is even and ample, then h ^ A , L h ^ A , L hat(h)_(A,L)\hat{h}_{A, \mathscr{L}}h^A,L takes nonnegative values and vanishes precisely on A tors A tors  A_("tors ")A_{\text {tors }}Ators .
(v) If L L L\mathscr{L}L is even and ample, then h ^ A , L h ^ A , L hat(h)_(A,L)\hat{h}_{A, \mathscr{L}}h^A,L induces a well-defined map A ( Q ¯ ) R A ( Q ¯ ) ⊗ R → A( bar(Q))oxRrarrA(\overline{\mathbb{Q}}) \otimes \mathbb{R} \rightarrowA(Q¯)⊗R→ [ 0 , ) [ 0 , ∞ ) [0,oo)[0, \infty)[0,∞). It is the square of a norm ∥ â‹… ∥ ||*||\|\cdot\|∥⋅∥ on the R R R\mathbb{R}R-vector space A ( Q ¯ ) R A ( Q ¯ ) ⊗ R A( bar(Q))oxRA(\overline{\mathbb{Q}}) \otimes \mathbb{R}A(Q¯)⊗R and satisfies the parallelogram equality.
The norm ∥ â‹… ∥ ||*||\|\cdot\|∥⋅∥ allows us to do geometry in the R R R\mathbb{R}R-vector space A ( Q ¯ ) R A ( Q ¯ ) ⊗ R A( bar(Q))oxRA(\overline{\mathbb{Q}}) \otimes \mathbb{R}A(Q¯)⊗R (which is infinite dimensional if dim A 1 ) dim ⁡ A ≥ 1 ) dim A >= 1)\operatorname{dim} A \geq 1)dim⁡A≥1). Indeed, for z , w A ( Q ¯ ) R z , w ∈ A ( Q ¯ ) ⊗ R z,w in A( bar(Q))oxRz, w \in A(\overline{\mathbb{Q}}) \otimes \mathbb{R}z,w∈A(Q¯)⊗R, we define
P , Q = 1 2 ( P + Q 2 P 2 Q 2 ) ⟨ P , Q ⟩ = 1 2 ∥ P + Q ∥ 2 − ∥ P ∥ 2 − ∥ Q ∥ 2 (:P,Q:)=(1)/(2)(||P+Q||^(2)-||P||^(2)-||Q||^(2))\langle P, Q\rangle=\frac{1}{2}\left(\|P+Q\|^{2}-\|P\|^{2}-\|Q\|^{2}\right)⟨P,Q⟩=12(∥P+Q∥2−∥P∥2−∥Q∥2)
Then , ⟨ ⋅ , ⋅ ⟩ (:*,*:)\langle\cdot, \cdot\rangle⟨⋅,⋅⟩ is a positive definite, symmetric, bilinear form.
By abuse of notation, we also write P ∥ P ∥ ||P||\|P\|∥P∥ and P , Q ⟨ P , Q ⟩ (:P,Q:)\langle P, Q\rangle⟨P,Q⟩ for P , Q A ( Q ¯ ) P , Q ∈ A ( Q ¯ ) P,Q in A( bar(Q))P, Q \in A(\overline{\mathbb{Q}})P,Q∈A(Q¯). In this notation we have P , P = h ^ A , L ( P ) ⟨ P , P ⟩ = h ^ A , L ( P ) (:P,P:)= hat(h)_(A,L)(P)\langle P, P\rangle=\hat{h}_{A, \mathscr{L}}(P)⟨P,P⟩=h^A,L(P).
The Mordell-Weil Theorem implies that A ( F ) R A ( F ) ⊗ R A(F)oxRA(F) \otimes \mathbb{R}A(F)⊗R is finite dimensional. We will see that ∥ ⋅ ∥ ||*||\|\cdot\|∥⋅∥ is a suitable norm to do Euclidean geometry in A ( F ) R A ( F ) ⊗ R A(F)oxRA(F) \otimes \mathbb{R}A(F)⊗R.

3. VOJTA'S APPROACH TO THE MORDELL CONJECTURE

Recall that the Mordell Conjecture was proved first by Faltings. In this section we briefly describe Vojta's approach to the Mordell Conjecture [62]. At the core is the deep Vojta inequality which we state here for a curve in an abelian variety.
Let A A AAA be an abelian variety defined over a number field F Q ¯ F ⊆ Q ¯ F sube bar(Q)F \subseteq \overline{\mathbb{Q}}F⊆Q¯. Let L L L\mathscr{L}L be an ample and even invertible sheaf on A A AAA. We write = h ^ A , L 1 / 2 ∥ â‹… ∥ = h ^ A , L 1 / 2 ||*||= hat(h)_(A,L)^(1//2)\|\cdot\|=\hat{h}_{A, \mathscr{L}}^{1 / 2}∥⋅∥=h^A,L1/2 for the norm on A ( Q ¯ ) R A ( Q ¯ ) ⊗ R A( bar(Q))oxRA(\overline{\mathbb{Q}}) \otimes \mathbb{R}A(Q¯)⊗R defined in Theorem 2.7.
Theorem 3.1 (Vojta's inequality). Let C A C ⊆ A C sube AC \subseteq AC⊆A be a curve that is defined over F F FFF and that is not a translate of an algebraic subgroup of A. There are c 1 > 1 , c 2 > 1 c 1 > 1 , c 2 > 1 c_(1) > 1,c_(2) > 1c_{1}>1, c_{2}>1c1>1,c2>1, and c 3 > 0 c 3 > 0 c_(3) > 0c_{3}>0c3>0 with the following property. If P , Q C ( Q ¯ ) P , Q ∈ C ( Q ¯ ) P,Q in C( bar(Q))P, Q \in C(\overline{\mathbb{Q}})P,Q∈C(Q¯) satisfy
P , Q ( 1 1 c 1 ) P Q ⟨ P , Q ⟩ ≥ 1 − 1 c 1 ∥ P ∥ ∥ Q ∥ (:P,Q:) >= (1-(1)/(c_(1)))||P||||Q||\langle P, Q\rangle \geq\left(1-\frac{1}{c_{1}}\right)\|P\|\|Q\|⟨P,Q⟩≥(1−1c1)∥P∥∥Q∥
and
Q c 2 P ∥ Q ∥ ≥ c 2 ∥ P ∥ ||Q|| >= c_(2)||P||\|Q\| \geq c_{2}\|P\|∥Q∥≥c2∥P∥
then P c 3 ∥ P ∥ ≤ c 3 ||P|| <= c_(3)\|P\| \leq c_{3}∥P∥≤c3.
We refer also to Rémond's work [55] for a completely explicit version of Vojta's inequality.
The values c 1 , c 2 , c 3 c 1 , c 2 , c 3 c_(1),c_(2),c_(3)c_{1}, c_{2}, c_{3}c1,c2,c3 depend on the curve C C CCC. One remarkable aspect is that Vojta's inequality is a statement about pairs of Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯-points of the curve C C CCC. So all three values c 1 , c 2 , c 3 c 1 , c 2 , c 3 c_(1),c_(2),c_(3)c_{1}, c_{2}, c_{3}c1,c2,c3 are "absolute," i.e., we can take them as independent of the base field F F FFF of A A AAA and C C CCC. Both c 1 c 1 c_(1)c_{1}c1 and c 2 c 2 c_(2)c_{2}c2 are of "geometric nature." They depend only on the degree of C C CCC with respect to L L L\mathscr{L}L and other discrete data attached to A A AAA and C C CCC. In contrast, c 3 c 3 c_(3)c_{3}c3 is of "arithmetic nature." Roughly speaking, it depends on suitable heights of coefficients that define the curve C C CCC in some projective embedding.
Let us now sketch a proof of Mordell's Conjecture using the Vojta inequality and the classical Mordell-Weil Theorem.
Suppose C C CCC has genus g 2 g ≥ 2 g >= 2g \geq 2g≥2. Without loss of generality, C ( F ) C ( F ) ≠ ∅ C(F)!=O/C(F) \neq \emptysetC(F)≠∅. So we fix a base point P 0 C ( F ) P 0 ∈ C ( F ) P_(0)in C(F)P_{0} \in C(F)P0∈C(F), then P P P 0 P ↦ P − P 0 P|->P-P_(0)P \mapsto P-P_{0}P↦P−P0 induces an immersion C Jac ( C ) C → Jac ⁡ ( C ) C rarr Jac(C)C \rightarrow \operatorname{Jac}(C)C→Jac⁡(C). So we may assume that C C CCC is a curve inside the g g ggg-dimensional A = Jac ( C ) A = Jac ⁡ ( C ) A=Jac(C)A=\operatorname{Jac}(C)A=Jac⁡(C). Note that C C CCC is not a translate of an algebraic subgroup of its Jacobian since g 2 g ≥ 2 g >= 2g \geq 2g≥2.
We observe that C ( F ) = C ( Q ¯ ) Jac ( C ) ( F ) C ( F ) = C ( Q ¯ ) ∩ Jac ⁡ ( C ) ( F ) C(F)=C( bar(Q))nn Jac(C)(F)C(F)=C(\overline{\mathbb{Q}}) \cap \operatorname{Jac}(C)(F)C(F)=C(Q¯)∩Jac⁡(C)(F).
By the Northcott property, stated below Theorem 2.4, combined with Theorem 2.7(ii) we find that the "ball"
(3.1) { P C ( F ) : P 2 B } (3.1) P ∈ C ( F ) : ∥ P ∥ 2 ≤ B {:(3.1){P in C(F):||P||^(2) <= B}:}\begin{equation*} \left\{P \in C(F):\|P\|^{2} \leq B\right\} \tag{3.1} \end{equation*}(3.1){P∈C(F):∥P∥2≤B}
is finite for all B B BBB.
We split the set of points C ( F ) C ( F ) C(F)C(F)C(F) into two subsets:
{ P C ( F ) : P 2 > c 3 } (large points), { P C ( F ) : P 2 c 3 } (moderate points). P ∈ C ( F ) : ∥ P ∥ 2 > c 3  (large points),  P ∈ C ( F ) : ∥ P ∥ 2 ≤ c 3  (moderate points).  {:[{P in C(F):||P||^(2) > c_(3)}," (large points), "],[{P in C(F):||P||^(2) <= c_(3)}," (moderate points). "]:}\begin{array}{ll} \left\{P \in C(F):\|P\|^{2}>c_{3}\right\} & \text { (large points), } \\ \left\{P \in C(F):\|P\|^{2} \leq c_{3}\right\} & \text { (moderate points). } \end{array}{P∈C(F):∥P∥2>c3} (large points), {P∈C(F):∥P∥2≤c3} (moderate points). 
By the finiteness statement around (3.1), it suffices to show that there are at most finitely many large points.
For any z Jac ( C ) ( F ) R z ∈ Jac ⁡ ( C ) ( F ) ⊗ R z in Jac(C)(F)oxRz \in \operatorname{Jac}(C)(F) \otimes \mathbb{R}z∈Jac⁡(C)(F)⊗R, we define the truncated cone
T ( z ) = { w Jac ( C ) ( F ) R : z , w ( 1 1 / c 1 ) z w and w 2 > c 3 } Jac ( C ) ( F ) R T ( z ) = w ∈ Jac ⁡ ( C ) ( F ) ⊗ R : ⟨ z , w ⟩ ≥ 1 − 1 / c 1 ∥ z ∥ ∥ w ∥  and  ∥ w ∥ 2 > c 3 ⊆ Jac ⁡ ( C ) ( F ) ⊗ R {:[T(z)={w in Jac(C)(F)oxR:(:z,w:) >= (1-1//c_(1))||z||||w||" and "||w||^(2) > c_(3)}],[ sube Jac(C)(F)oxR]:}\begin{aligned} T(z) & =\left\{w \in \operatorname{Jac}(C)(F) \otimes \mathbb{R}:\langle z, w\rangle \geq\left(1-1 / c_{1}\right)\|z\|\|w\| \text { and }\|w\|^{2}>c_{3}\right\} \\ & \subseteq \operatorname{Jac}(C)(F) \otimes \mathbb{R} \end{aligned}T(z)={w∈Jac⁡(C)(F)⊗R:⟨z,w⟩≥(1−1/c1)∥z∥∥w∥ and ∥w∥2>c3}⊆Jac⁡(C)(F)⊗R
By the Mordell-Weil Theorem, Jac ( C ) ( F ) R Jac ⁡ ( C ) ( F ) ⊗ R Jac(C)(F)oxR\operatorname{Jac}(C)(F) \otimes \mathbb{R}Jac⁡(C)(F)⊗R is a finite-dimensional R R R\mathbb{R}R-vector space. So the unit sphere with respect to the norm ∥ ⋅ ∥ ||*||\|\cdot\|∥⋅∥ coming from the Néron-Tate height
is compact. Therefore, { w Jac ( C ) ( F ) R : w 2 > c 3 } w ∈ Jac ⁡ ( C ) ( F ) ⊗ R : ∥ w ∥ 2 > c 3 {w in Jac(C)(F)oxR:||w||^(2) > c_(3)}\left\{w \in \operatorname{Jac}(C)(F) \otimes \mathbb{R}:\|w\|^{2}>c_{3}\right\}{w∈Jac⁡(C)(F)⊗R:∥w∥2>c3} is covered by a finite union T ( z 1 ) T ( z N ) T z 1 ∪ ⋯ ∪ T z N T(z_(1))uu cdots uu T(z_(N))T\left(z_{1}\right) \cup \cdots \cup T\left(z_{N}\right)T(z1)∪⋯∪T(zN). Using a sphere packing argument, one can arrange that N N NNN is bounded from above by c c r k J a c ( C ) ( F ) c ′ ⋅ c r k J a c ( C ) ( F ) c^(')*c^(rkJac(C)(F))c^{\prime} \cdot c^{\mathrm{rkJac}(C)(F)}c′⋅crkJac(C)(F) where c > 0 c ′ > 0 c^(') > 0c^{\prime}>0c′>0 and c > 1 c > 1 c > 1c>1c>1 depend only on c 1 c 1 c_(1)c_{1}c1. This observation will be important for deriving uniform bounds for # C ( F ) # C ( F ) #C(F)\# C(F)#C(F).
Any large point in C ( F ) C ( F ) C(F)C(F)C(F) has image in some T ( z j ) T z j T(z_(j))T\left(z_{j}\right)T(zj) from above. After possibly adjusting N N NNN, one can arrange that each z j z j z_(j)z_{j}zj is the image of a point P j C ( F ) P j ∈ C ( F ) P_(j)in C(F)P_{j} \in C(F)Pj∈C(F) with P j 2 > c 3 P j 2 > c 3 ||P_(j)||^(2) > c_(3)\left\|P_{j}\right\|^{2}>c_{3}∥Pj∥2>c3 for all j { 1 , , N } j ∈ { 1 , … , N } j in{1,dots,N}j \in\{1, \ldots, N\}j∈{1,…,N}. If Q C ( F ) Q ∈ C ( F ) Q in C(F)Q \in C(F)Q∈C(F) has image in T ( z j ) T z j T(z_(j))T\left(z_{j}\right)T(zj), then Vojta's inequality implies h ^ L ( Q ) 1 / 2 = Q c 2 P j h ^ L ( Q ) 1 / 2 = ∥ Q ∥ ≤ c 2 P j hat(h)_(L)(Q)^(1//2)=||Q|| <= c_(2)||P_(j)||\hat{h}_{\mathscr{L}}(Q)^{1 / 2}=\|Q\| \leq c_{2}\left\|P_{j}\right\|h^L(Q)1/2=∥Q∥≤c2∥Pj∥. But then Q C ( F ) Q ∈ C ( F ) Q in C(F)Q \in C(F)Q∈C(F) lies in a finite ball as in (3.1). So the number of possible Q Q QQQ that come to lie in a single T ( z j ) T z j T(z_(j))T\left(z_{j}\right)T(zj) is finite. Thus C ( F ) C ( F ) C(F)C(F)C(F) is finite.
The constants c 1 , c 2 c 1 , c 2 c_(1),c_(2)c_{1}, c_{2}c1,c2, and c 3 c 3 c_(3)c_{3}c3 in Vojta's inequality can be made effective in terms of A A AAA and C C CCC. Yet, the proof as a whole is ineffective. Indeed, the height bound for Q Q QQQ depends on the hypothetical point P j P j P_(j)P_{j}Pj. However, there is no guarantee that P j P j P_(j)P_{j}Pj exists and if it does not, there is no known way to know for sure.
Using Mumford's Gap Principle, one can show that the number of large points C ( F ) C ( F ) C(F)C(F)C(F) that come to lie in a single T ( z j ) T z j T(z_(j))T\left(z_{j}\right)T(zj) is bounded from above by c c r k J a c ( C ) ( F ) c ′ ⋅ c r k J a c ( C ) ( F ) c^(')*c^(rkJac(C)(F))c^{\prime} \cdot c^{\mathrm{rkJac}(C)(F)}c′⋅crkJac(C)(F), after possibly increasing the constants. Now we need to introduce dependency on c 2 c 2 c_(2)c_{2}c2. But the base c c ccc will remain geometric in nature, it depends on the genus of g g ggg. But it does not depend on c 3 c 3 c_(3)c_{3}c3 or other arithmetic properties of C C CCC that encode the heights of coefficients defining the said curve. Finally, as observed by Bombieri, 7 is admissible for c c ccc for any genus. Indeed, he showed that 4 is admissible for c 1 c 1 c_(1)c_{1}c1.
Recall that Vojta's inequality with the same values of c 1 , c 2 , c 3 c 1 , c 2 , c 3 c_(1),c_(2),c_(3)c_{1}, c_{2}, c_{3}c1,c2,c3 applies to points in C ( F ) C F ′ C(F^('))C\left(F^{\prime}\right)C(F′) for all finite extensions F / F F ′ / F F^(')//FF^{\prime} / FF′/F. The upshot is that the number of large points of C ( F ) C F ′ C(F^('))C\left(F^{\prime}\right)C(F′) is bounded by
c c r k J a c ( C ) ( F ) c ′ ⋅ c r k J a c ( C ) F ′ c^(')*c^(rkJac(C)(F^(')))c^{\prime} \cdot c^{\mathrm{rkJac}(C)\left(F^{\prime}\right)}c′⋅crkJac(C)(F′)
where c , c c , c ′ c,c^(')c, c^{\prime}c,c′ depend on C C CCC, but not on F F ′ F^(')F^{\prime}F′.
The dichotomy between large and moderate points was already visible in Vojta's work. But its origin is older and already appears in modified form in work of Thue, Siegel, Mahler, and Roth on diophantine approximation.
Rémond's explicit Théorème 2.1 [54] gives a recipe how to bound the total number of rational points using a bound for the number of moderate points.
With our eyes set on Mazur's question, we aim to obtain good bounds for the number of moderate points. In the coming two sections we explain our general approach to the proof of Theorem 1.12.

4. COMPARING WEIL AND NÉRON-TATE HEIGHTS

The interplay between the Weil and Néron-Tate heights on a family of abelian varieties leads to powerful results including Silverman's Specialization Theorem [57] and more recent work by Masser and Zannier towards the relative Manin-Mumford Conjecture [44]. This interaction also plays a central role in the proof of Theorem 1.12 that resolved Mazur's question.
Having worked with a fixed abelian variety in Sections 2.2 and 3, we now shift gears and work in a family of abelian varieties.
Example 4.1. Let Y ( 2 ) = P 1 { 0 , 1 , } Y ( 2 ) = P 1 ∖ { 0 , 1 , ∞ } Y(2)=P^(1)\\{0,1,oo}Y(2)=\mathbb{P}^{1} \backslash\{0,1, \infty\}Y(2)=P1∖{0,1,∞}. For λ Y ( 2 ) ( C ) λ ∈ Y ( 2 ) ( C ) lambda in Y(2)(C)\lambda \in Y(2)(\mathbb{C})λ∈Y(2)(C), we have an elliptic curve E λ P 2 E λ ⊆ P 2 E_(lambda)subeP^(2)\mathcal{E}_{\lambda} \subseteq \mathbb{P}^{2}Eλ⊆P2 determined by
y 2 z = x ( x z ) ( x λ z ) y 2 z = x ( x − z ) ( x − λ z ) y^(2)z=x(x-z)(x-lambda z)y^{2} z=x(x-z)(x-\lambda z)y2z=x(x−z)(x−λz)
where the origin is [ 0 : 1 : 0 ] [ 0 : 1 : 0 ] [0:1:0][0: 1: 0][0:1:0]. The total space E E E\mathcal{E}E is a surface presented with a closed immersion E P 2 × Y ( 2 ) E ↪ P 2 × Y ( 2 ) E↪P^(2)xx Y(2)\mathcal{E} \hookrightarrow \mathbb{P}^{2} \times Y(2)E↪P2×Y(2). It is called the Legendre family of elliptic curves and is an abelian scheme over Y ( 2 ) Y ( 2 ) Y(2)Y(2)Y(2). So we can add two complex points of E E E\mathscr{E}E if they are in the same fiber above Y ( 2 ) Y ( 2 ) Y(2)Y(2)Y(2). More precisely, there is an addition morphism E × S E E E × S E → E Exx_(S)ErarrE\mathcal{E} \times_{S} \mathcal{E} \rightarrow \mathcal{E}E×SE→E over S S SSS, as well as an inversion morphism E E E → E ErarrE\mathcal{E} \rightarrow \mathcal{E}E→E over S S SSS. Finally, the zero section of E E E\mathcal{E}E is given by λ ( [ 0 : 1 : 0 ] , λ ) λ ↦ ( [ 0 : 1 : 0 ] , λ ) lambda|->([0:1:0],lambda)\lambda \mapsto([0: 1: 0], \lambda)λ↦([0:1:0],λ).
Consider a geometrically irreducible smooth quasiprojective variety S S SSS defined over a number field F Q ¯ F ⊆ Q ¯ F sube bar(Q)F \subseteq \overline{\mathbb{Q}}F⊆Q¯. Let π : A S Ï€ : A → S pi:Ararr S\pi: \mathcal{A} \rightarrow SÏ€:A→S be an abelian scheme over S S SSS. So each fiber A s = π 1 ( s ) A s = Ï€ − 1 ( s ) A_(s)=pi^(-1)(s)\mathcal{A}_{s}=\pi^{-1}(s)As=π−1(s), where s S ( Q ¯ ) s ∈ S ( Q ¯ ) s in S( bar(Q))s \in S(\overline{\mathbb{Q}})s∈S(Q¯), is an abelian variety. We have an addition morphism on the fibered square A × × S A A A × × S A → A Axxxx_(S)ArarrA\mathcal{A} \times \times_{S} \mathcal{A} \rightarrow \mathcal{A}A××SA→A and an inversion morphism A A A → A ArarrA\mathscr{A} \rightarrow \mathcal{A}A→A; both are relative over S S SSS. Addition induces a multiplication-by- n n nnn morphism [ n ] : A A [ n ] : A → A [n]:ArarrA[n]: \mathcal{A} \rightarrow \mathcal{A}[n]:A→A over S S SSS for all n Z n ∈ Z n inZn \in \mathbb{Z}n∈Z.
For simplicity, we assume that A A A\mathscr{A}A is presented with an immersion A P n × S A ↪ P n × S A↪P^(n)xx S\mathscr{A} \hookrightarrow \mathbb{P}^{n} \times SA↪Pn×S over S S SSS, much as in Example 4.1 above. Let L L L\mathscr{L}L be the restriction of the hyperplane bundle O ( 1 ) O ( 1 ) O(1)\mathcal{O}(1)O(1) on P n × S = P S n P n × S = P S n P^(n)xx S=P_(S)^(n)\mathbb{P}^{n} \times S=\mathbb{P}_{S}^{n}Pn×S=PSn to A A A\mathscr{A}A. We also assume that L L L\mathscr{L}L is even, that is [ 1 ] L L [ − 1 ] ∗ L ≅ L [-1]^(**)L~=L[-1]^{*} \mathscr{L} \cong \mathscr{L}[−1]∗L≅L. This allows us to define a fiberwise Néron-Tate height on A ( Q ¯ ) A ( Q ¯ ) A( bar(Q))\mathcal{A}(\overline{\mathbb{Q}})A(Q¯) which we abbreviate by h ^ A h ^ A hat(h)_(A)\hat{h}_{\mathcal{A}}h^A.
Let s S ( Q ¯ ) s ∈ S ( Q ¯ ) s in S( bar(Q))s \in S(\overline{\mathbb{Q}})s∈S(Q¯). Then A s A s A_(s)\mathscr{A}_{s}As is an abelian variety in P n P n P^(n)\mathbb{P}^{n}Pn. We have two functions, h ^ A | A s ( Q ¯ ) h ^ A A s ( Q ¯ ) hat(h)_(A)|_(A_(s)( bar(Q)))\left.\hat{h}_{\mathcal{A}}\right|_{\mathcal{A}_{s}(\overline{\mathbb{Q}})}h^A|As(Q¯) and h | A s ( Q ¯ ) h A s ( Q ¯ ) h|_(A_(s)( bar(Q)))\left.h\right|_{\mathcal{A}_{s}(\overline{\mathbb{Q}})}h|As(Q¯); the latter is the restriction of the Weil height on P n P n P^(n)\mathbb{P}^{n}Pn. By Theorem 2.7(ii), their difference is bounded in absolute value in function of s s sss.
In the example of the Legendre family, the point [ λ : 0 : 1 ] E λ [ λ : 0 : 1 ] ∈ E λ [lambda:0:1]inE_(lambda)[\lambda: 0: 1] \in \mathcal{E}_{\lambda}[λ:0:1]∈Eλ is of order 2 for all λ λ lambda\lambdaλ. So its Néron-Tate height vanishes, but its Weil height equals h ( [ λ : 1 ] ) h ( [ λ : 1 ] ) h([lambda:1])h([\lambda: 1])h([λ:1]) and is thus unbounded as λ λ lambda\lambdaλ varies.
We would like to understand the difference between Néron-Tate and Weil heights on A A A\mathcal{A}A as the base point s S ( Q ¯ ) s ∈ S ( Q ¯ ) s in S( bar(Q))s \in S(\overline{\mathbb{Q}})s∈S(Q¯) varies. As suggested by the Legendre case, the key is the Weil height on the base S S SSS. To keep things concrete, we will assume that S S SSS comes with an immersion S P m S ↪ P m S↪P^(m)S \hookrightarrow \mathbb{P}^{m}S↪Pm. We identify S S SSS with a Zariski locally closed subset of P m P m P^(m)\mathbb{P}^{m}Pm. So S S SSS need not be projective, but its Zariski closure S ¯ S ¯ bar(S)\bar{S}S¯ in P m P m P^(m)\mathbb{P}^{m}Pm is. We write h S h S h_(S)h_{S}hS for h | S ¯ ( Q ¯ ) : S ¯ ( Q ¯ ) [ 0 , ) h S ¯ ( Q ¯ ) : S ¯ ( Q ¯ ) → [ 0 , ∞ ) h|_( bar(S)( bar(Q))): bar(S)( bar(Q))rarr[0,oo)\left.h\right|_{\bar{S}(\overline{\mathbb{Q}})}: \bar{S}(\overline{\mathbb{Q}}) \rightarrow[0, \infty)h|S¯(Q¯):S¯(Q¯)→[0,∞) where h h hhh is the Weil height on P m ( Q ¯ ) P m ( Q ¯ ) P^(m)( bar(Q))\mathbb{P}^{m}(\overline{\mathbb{Q}})Pm(Q¯). In the language of Section 2.1, h S h S h_(S)h_{S}hS represents the Weil height attached to ( S ¯ , O ( 1 ) | S ¯ ) S ¯ , O ( 1 ) S ¯ (( bar(S)),O(1)|_( bar(S)))\left(\bar{S},\left.\mathcal{O}(1)\right|_{\bar{S}}\right)(S¯,O(1)|S¯).
The difference between Weil and Néron-Tate heights on the total space A ( Q ¯ ) A ( Q ¯ ) A( bar(Q))\mathcal{A}(\overline{\mathbb{Q}})A(Q¯) was clarified in work of Zimmer [73] in the elliptic setting and Manin-Zarhin [69] and SilvermanTate [57] in the more general setting. In our case the latter result amounts to
for all s S ( Q ¯ ) s ∈ S ( Q ¯ ) s in S( bar(Q))s \in S(\overline{\mathbb{Q}})s∈S(Q¯).
We introduce a final player, a geometrically-irreducible subvariety V V VVV of A A A\mathcal{A}A defined over F F FFF.
Theorem 4.2 (Silverman [57]). Suppose S S SSS and V A V ⊆ A V subeAV \subseteq \mathcal{A}V⊆A are curves such that V V VVV dominates S S SSS. Then
(4.2) lim P V ( Q ¯ ) h ( π ( P ) ) h ^ A ( P ) h ( π ( P ) ) (4.2) lim P ∈ V ( Q ¯ ) h ( Ï€ ( P ) ) → ∞   h ^ A ( P ) h ( Ï€ ( P ) ) {:(4.2)lim_({:[P in V( bar(Q))],[h(pi(P))rarr oo]:})( hat(h)_(A)(P))/(h(pi(P))):}\begin{equation*} \lim _{\substack{P \in V(\overline{\mathbb{Q}}) \\ h(\pi(P)) \rightarrow \infty}} \frac{\hat{h}_{\mathcal{A}}(P)}{h(\pi(P))} \tag{4.2} \end{equation*}(4.2)limP∈V(Q¯)h(Ï€(P))→∞h^A(P)h(Ï€(P))
exists. Suppose, in addition, that the geometric generic fiber of A S A → S Ararr S\mathcal{A} \rightarrow SA→S has trivial trace over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯. Then the limit vanishes if and only if V V VVV is an irreducible component of ker [ N ] ker ⁡ [ N ] ker[N]\operatorname{ker}[N]ker⁡[N] for some N 1 N ≥ 1 N >= 1N \geq 1N≥1. Otherwise the limit is positive.
Silverman computed the limit in terms of the Néron-Tate height of V V VVV restricted to the generic fiber A S A → S Ararr S\mathscr{A} \rightarrow SA→S.
The "if direction" is straightforward: in this case all P P PPP in question are of finite order and their Néron-Tate height vanishes; see Theorem 2.7(iv). The "only if" direction is deeper and has many applications: Silverman's Specialization Theorem, Theorem C in [57], as well as applications to unlikely intersections by Masser and Zannier, see [44] and [68] for an overview and more results.
What happens if V V VVV has dimension > 1 > 1 > 1>1>1 and S S SSS remains a curve? In this case the limit (4.2) does not make sense. Indeed, for fixed s S ( Q ¯ ) s ∈ S ( Q ¯ ) s in S( bar(Q))s \in S(\overline{\mathbb{Q}})s∈S(Q¯), the set of P V ( Q ¯ ) P ∈ V ( Q ¯ ) P in V( bar(Q))P \in V(\overline{\mathbb{Q}})P∈V(Q¯) that map to s s sss has positive dimension and thus unbounded Néron-Tate height.
Motivated by Theorem 4.2, the author showed the next theorem. It may serve as a higher-dimensional substitute for Silverman's Theorem 4.2. For an irreducible subvariety of V V VVV of A A A\mathcal{A}A that dominates S S SSS, we write V η ¯ V η ¯ V_( bar(eta))V_{\bar{\eta}}Vη¯ for the geometric generic fiber of π | V : V S Ï€ V : V → S pi|_(V):V rarr S\left.\pi\right|_{V}: V \rightarrow SÏ€|V:V→S. This is a possibly reducible subvariety of the geometric generic fiber A η ¯ A η ¯ A_( bar(eta))\mathcal{A}_{\bar{\eta}}Aη¯ of A S A → S Ararr S\mathcal{A} \rightarrow SA→S.
Theorem 4.3 ([35]). Suppose S = Y ( 2 ) S = Y ( 2 ) S=Y(2)S=Y(2)S=Y(2) and let A = E [ g ] A = E [ g ] A=E[g]\mathcal{A}=\mathcal{E}[g]A=E[g] be the g g ggg-fold fibered power of the Legendre family of elliptic curves. Suppose V E [ g ] V ⊆ E [ g ] V subeE^([g])V \subseteq \mathcal{E}^{[g]}V⊆E[g] dominates Y ( 2 ) Y ( 2 ) Y(2)Y(2)Y(2) and
V η ¯ V η ¯ V_( bar(eta))V_{\bar{\eta}}Vη¯ is not a finite union of irreducible components of algebraic subgroups of A η ¯ A η ¯ A_( bar(eta))\mathcal{A}_{\bar{\eta}}Aη¯.
Then there exist c ( V ) > 0 c ( V ) > 0 c(V) > 0c(V)>0c(V)>0 and a Zariski open and dense subset U V U ⊆ V U sube VU \subseteq VU⊆V with
(4.4) h Y ( 2 ) ( π ( P ) ) c ( V ) max { 1 , h ^ A ( P ) } for all P U ( Q ¯ ) (4.4) h Y ( 2 ) ( Ï€ ( P ) ) ≤ c ( V ) max 1 , h ^ A ( P )  for all  P ∈ U ( Q ¯ ) {:(4.4)h_(Y(2))(pi(P)) <= c(V)max{1, hat(h)_(A)(P)}quad" for all "P in U( bar(Q)):}\begin{equation*} h_{Y(2)}(\pi(P)) \leq c(V) \max \left\{1, \hat{h}_{\mathcal{A}}(P)\right\} \quad \text { for all } P \in U(\overline{\mathbb{Q}}) \tag{4.4} \end{equation*}(4.4)hY(2)(Ï€(P))≤c(V)max{1,h^A(P)} for all P∈U(Q¯)
Say (4.3) holds. If P U ( Q ¯ ) P ∈ U ( Q ¯ ) P in U( bar(Q))P \in U(\overline{\mathbb{Q}})P∈U(Q¯) has finite order as a point in its respective fiber, we find h Y ( 2 ) ( π ( p ) ) c ( V ) h Y ( 2 ) ( Ï€ ( p ) ) ≤ c ( V ) h_(Y(2))(pi(p)) <= c(V)h_{Y(2)}(\pi(p)) \leq c(V)hY(2)(Ï€(p))≤c(V) and the total Weil height of P P PPP is bounded from above by (4.1). This simple observation led to the resolution of several "special points" problems [35] in the spirit of the André-Oort Conjecture. For example, torsion points P V ( Q ¯ ) P ∈ V ( Q ¯ ) P in V( bar(Q))P \in V(\overline{\mathbb{Q}})P∈V(Q¯) that lie in a fiber with complex multiplication are not Zariski dense in V V VVV. The proof of Theorem 4.3 makes use of Siu's Criterion, see Remark 2.5(iii), and an investigation of monodromy in E [ g ] E [ g ] E^([g])\mathcal{E}^{[g]}E[g].
The Zariski open U U UUU cannot in general be taken to equal V V VVV. But there is a natural description of this set in geometric terms through unlikely intersections.
The hypothesis (4.3) is necessary and essentially rules out that V V VVV itself is a family of abelian subvarieties.
Gao and the author [33] then generalized Theorem 4.3 to an abelian scheme when the base is again a smooth curve S S SSS defined over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯. Here more care is needed in connection
with the hypothesis (4.3). Indeed, if A = A × S A = A × S A=A xx S\mathcal{A}=A \times SA=A×S is a constant abelian scheme, where A A AAA is an abelian variety, then (4.4) cannot hold generically for V = Y × S V = Y × S V=Y xx SV=Y \times SV=Y×S. Roughly speaking, the condition in [33] that replaces (4.3) also needs to take into account a possible constant part of A η ¯ A η ¯ A_( bar(eta))\mathscr{A}_{\bar{\eta}}Aη¯. If A η ¯ A η ¯ A_( bar(eta))\mathscr{A}_{\bar{\eta}}Aη¯ has no constant part, i.e., if its Q ¯ ( η ¯ ) / Q ¯ Q ¯ ( η ¯ ) / Q ¯ bar(Q)( bar(eta))// bar(Q)\overline{\mathbb{Q}}(\bar{\eta}) / \overline{\mathbb{Q}}Q¯(η¯)/Q¯-trace is 0 , then (4.3) suffices for S S SSS a curve. The case of a higher-dimensional base requires even more care, as we will see.
There were two applications of the height bound in [33].
First, and in the same paper, we proved new cases of the geometric Bogomolov Conjecture for an abelian variety defined over the function field of the curve S S SSS. This approach relied on Silverman's Theorem 4.2. It was used earlier in [35] to give a new proof of the Geometric Bogomolov Conjecture in a power of an elliptic curve. The number field case of the Bogomolov Conjecture was proved by Ullmo [61] and Zhang [71] in the 1990s. Progress in the function field case was later made by Cinkir, Faber, Moriwaki, Gubler, and Yamaki. For the state of the Geometric Bogomolov Conjecture as of 2017, we refer to a survey of Yamaki [64]. Gubler's strategy works in arbitrary characteristic and was expanded on by Yamaki. In joint work [10] with Cantat, Gao, and Xie, the author later established the Geometric Bogomolov Conjecture in characteristic 0 by bypassing the height inequality (4.4). Very recently, Xie and Yuan [63] announced a proof of the Geometric Bogomolov Conjecture in arbitrary characteristic. Their approach builds on the work of Gubler and Yamaki.
Second, and in later joint work with Dimitrov and Gao [23], we established uniformity for the number of rational points in the spirit of Mazur's question for curves parametrized by the 1-dimensional base S S SSS.
As we shall see, the proof of Theorem 1.12 requires a height comparison result like (4.4) for abelian schemes over a base S S SSS of any dimension. But now the correct condition to impose on V V VVV is more sophisticated and cannot be easily read off of the geometric generic fiber as in (4.3). The condition relies on the Betti map, which we introduce in the next section.

4.1. Degenerate subvarieties and the Betti map

In this section, S S SSS is a smooth irreducible quasiprojective variety over C C C\mathbb{C}C. Let π : A S Ï€ : A → S pi:Ararr S\pi: \mathcal{A} \rightarrow SÏ€:A→S again be an abelian scheme over S S SSS of relative dimension g 1 g ≥ 1 g >= 1g \geq 1g≥1.
For each s S ( C ) s ∈ S ( C ) s in S(C)s \in S(\mathbb{C})s∈S(C), the fiber A s ( C ) A s ( C ) A_(s)(C)\mathcal{A}_{s}(\mathbb{C})As(C) is a complex torus of dimension g g ggg. Forgetting the complex structure, each g g ggg-dimensional complex torus is diffeomorphic to ( R / Z ) 2 g ( R / Z ) 2 g (R//Z)^(2g)(\mathbb{R} / \mathbb{Z})^{2 g}(R/Z)2g as a real Lie group. By Ehresmann's Theorem, this diffeomorphism extends locally in the analytic topology on the base. That is, there is a contractible open neighborhood U U UUU of s s sss in S ( C ) S ( C ) S(C)S(\mathbb{C})S(C) and a diffeomorphism A U = π 1 ( U ) ( R / Z ) 2 g × U A U = Ï€ − 1 ( U ) → ( R / Z ) 2 g × U A_(U)=pi^(-1)(U)rarr(R//Z)^(2g)xx U\mathscr{A}_{U}=\pi^{-1}(U) \rightarrow(\mathbb{R} / \mathbb{Z})^{2 g} \times UAU=π−1(U)→(R/Z)2g×U over U U UUU. Fiberwise this diffeomorphism can be arranged to be a group isomorphism above each point of U U UUU. Thus we can locally trivialize the abelian scheme at the cost of sacrificing the complex-analytic structure.
The trivialization is not entirely unique as we can let a matrix in G L 2 g ( Z ) G L 2 g ( Z ) GL_(2g)(Z)\mathrm{GL}_{2 g}(\mathbb{Z})GL2g(Z) act in the natural way on the real torus ( R / Z ) 2 g ( R / Z ) 2 g (R//Z)^(2g)(\mathbb{R} / \mathbb{Z})^{2 g}(R/Z)2g. But since U U UUU is connected, this is the only ambiguity. It is harmless for what follows.
The Betti map β U β U beta_(U)\beta_{U}βU attached to U U UUU is the composition of the trivialization followed by the projection
This map has appeared implicitly in diophantine geometry in work of Masser and Zannier [44]. We also refer to more recent work of André, Corvaja, and Zannier [3] for a systematic study of the Betti map.
We list some of the most important properties:
(i) For all s U s ∈ U s in Us \in Us∈U, the restriction β U | A s ( C ) : A s ( C ) ( R / Z ) 2 g β U A s ( C ) : A s ( C ) → ( R / Z ) 2 g beta_(U)|_(A_(s)(C)):A_(s)(C)rarr(R//Z)^(2g)\left.\beta_{U}\right|_{\mathcal{A}_{s}(\mathbb{C})}: \mathscr{A}_{s}(\mathbb{C}) \rightarrow(\mathbb{R} / \mathbb{Z})^{2 g}βU|As(C):As(C)→(R/Z)2g is a diffeomorphism of real Lie groups. In particular, P A U P ∈ A U P inA_(U)P \in \mathcal{A}_{U}P∈AU has finite order in its respective fiber if and only if β U ( P ) ( Q / Z ) 2 g β U ( P ) ∈ ( Q / Z ) 2 g beta_(U)(P)in(Q//Z)^(2g)\beta_{U}(P) \in(\mathbb{Q} / \mathbb{Z})^{2 g}βU(P)∈(Q/Z)2g.
(ii) For all P U P ∈ U P in UP \in UP∈U the fiber β U 1 ( β U ( P ) ) β U − 1 β U ( P ) beta_(U)^(-1)(beta_(U)(P))\beta_{U}^{-1}\left(\beta_{U}(P)\right)βU−1(βU(P)) is a complex-analytic subset of A U A U A_(U)\mathcal{A}_{U}AU.
Definition 4.4. An irreducible closed subvariety V A V ⊆ A V subeAV \subseteq \mathcal{A}V⊆A that dominates S S SSS is called degenerate if for all U U UUU and β U β U beta_(U)\beta_{U}βU as above and all smooth points P P PPP of V U = π | V 1 ( U ) V U = Ï€ V − 1 ( U ) V_(U)= pi|_(V)^(-1)(U)V_{U}=\left.\pi\right|_{V} ^{-1}(U)VU=Ï€|V−1(U) the differential of d P ( β U | V U ) d P β U V U d_(P)(beta_(U)|_(V_(U)))\mathrm{d}_{P}\left(\left.\beta_{U}\right|_{V_{U}}\right)dP(βU|VU) satisfies
(4.5) rkd P ( β U | V U ) < 2 dim V (4.5) rkd P ⁡ β U V U < 2 dim ⁡ V {:(4.5)rkd_(P)(beta_(U)|_(V_(U))) < 2dim V:}\begin{equation*} \operatorname{rkd}_{P}\left(\left.\beta_{U}\right|_{V_{U}}\right)<2 \operatorname{dim} V \tag{4.5} \end{equation*}(4.5)rkdP⁡(βU|VU)<2dim⁡V
It has become customary to call V V VVV degenerate if it is not nondegenerate.
For all smooth points P P PPP of V U V U V_(U)V_{U}VU, the left-hand side of (4.5) is at most the right-hand side, which equals the real dimension of V U V U V_(U)V_{U}VU. It is also at most 2 g 2 g 2g2 g2g, the real dimension of a fiber of A S A → S Ararr S\mathcal{A} \rightarrow SA→S. Moreover, if the maximal rank of d β U d β U dbeta_(U)\mathrm{d} \beta_{U}dβU on V U V U V_(U)V_{U}VU is attained at P P PPP then the maximal rank is attained also in a neighborhood of P P PPP in V U V U V_(U)V_{U}VU. Being nondegenerate is a local property.
Let us consider some examples.
Example 4.5. (i) If S S SSS is a point, then A A A\mathscr{A}A is an abelian variety and an arbitrary subvariety V A V ⊆ A V subeAV \subseteq \mathcal{A}V⊆A is nondegenerate because β S β S beta_(S)\beta_{S}βS is a diffeomorphism.
(ii) Suppose dim V > g dim ⁡ V > g dim V > g\operatorname{dim} V>gdim⁡V>g. Then rkd P ( β U V U ) 2 g < 2 dim V rkd P ⁡ β U ∣ V U ≤ 2 g < 2 dim ⁡ V rkd_(P)(beta_(U)∣V_(U)) <= 2g < 2dim V\operatorname{rkd}_{P}\left(\beta_{U} \mid V_{U}\right) \leq 2 g<2 \operatorname{dim} VrkdP⁡(βU∣VU)≤2g<2dim⁡V for all smooth P P PPP and so V V VVV is degenerate. In particular, A A A\mathscr{A}A is a degenerate subvariety of A A A\mathscr{A}A if dim S 1 dim ⁡ S ≥ 1 dim S >= 1\operatorname{dim} S \geq 1dim⁡S≥1.
(iii) Suppose A = A × S A = A × S A=A xx S\mathscr{A}=A \times SA=A×S is a constant abelian scheme with A A AAA an abelian variety. If Y A Y ⊆ A Y sube AY \subseteq AY⊆A is a closed irreducible subvariety and if dim S 1 dim ⁡ S ≥ 1 dim S >= 1\operatorname{dim} S \geq 1dim⁡S≥1, then Y × S Y × S Y xx SY \times SY×S is degenerate. Indeed, the rank is at most 2 dim Y < 2 dim Y × S 2 dim ⁡ Y < 2 dim ⁡ Y × S 2dim Y < 2dim Y xx S2 \operatorname{dim} Y<2 \operatorname{dim} Y \times S2dim⁡Y<2dim⁡Y×S.
(iv) Suppose V V VVV is an irreducible component of ker [ N ] ker ⁡ [ N ] ker[N]\operatorname{ker}[N]ker⁡[N] for some integer N 1 N ≥ 1 N >= 1N \geq 1N≥1. Any point in V ( C ) V ( C ) V(C)V(\mathbb{C})V(C) has order dividing N N NNN (and, in fact, equal to N N NNN ). So the image of β U | V U β U V U beta_(U)|_(V_(U))\left.\beta_{U}\right|_{V_{U}}βU|VU is finite and hence V V VVV is degenerate if dim S 1 dim ⁡ S ≥ 1 dim S >= 1\operatorname{dim} S \geq 1dim⁡S≥1.
(v) Suppose V V VVV is the image of a section S A S → A S rarrAS \rightarrow \mathcal{A}S→A. If the geometric generic fiber of A S A → S Ararr S\mathcal{A} \rightarrow SA→S has trivial trace, then ( β U ) | V U β U V U (beta_(U))|_(V_(U))\left.\left(\beta_{U}\right)\right|_{V_{U}}(βU)|VU is constant if and only if V V VVV is an irreducible component of ker [ N ] ker ⁡ [ N ] ker[N]\operatorname{ker}[N]ker⁡[N] for some N 1 N ≥ 1 N >= 1N \geq 1N≥1. This is Manin's Theorem of the Kernel, we refer to Bertrand's article [7] for the history of this theorem.
(vi) Suppose A = E [ g ] A = E [ g ] A=E^([g])\mathcal{A}=\mathcal{E}^{[g]}A=E[g] and V V VVV are as in Theorem 4.3. One step in the proof of this theorem consisted in verifying that V V VVV, subject to hypothesis (4.3), is nondegenerate. Crucial input came from the monodromy action of the fundamental group of the base Y ( 2 ) = P 1 { 0 , 1 , } Y ( 2 ) = P 1 ∖ { 0 , 1 , ∞ } Y(2)=P^(1)\\{0,1,oo}Y(2)=\mathbb{P}^{1} \backslash\{0,1, \infty\}Y(2)=P1∖{0,1,∞} on the first homology of a fiber A s A s A_(s)\mathcal{A}_{s}As with s s sss in general position. In this case the monodromy action is unipotent at the cusps 0 and 1 of Y Y YYY (2). This enabled the author to use a result of Kronecker from diophantine approximation. Already Masser and Zannier [44] used the monodromy action in their earlier work for V V VVV a curve.
(vii) If S S SSS is a curve, then the monodromy action of the fundamental group of S ( C ) S ( C ) S(C)S(\mathbb{C})S(C) on the homology of fibers of A S A → S Ararr S\mathscr{A} \rightarrow SA→S is locally quasiunipotent. But if S S SSS is projective, then there are no cusps. So exploiting monodromy in this setting required a different approach. In [33] Gao and the author used o-minimal geometry and the Pila-Wilkie Counting Theorem [51]. A related case was solved by Cantat, Gao, and Xie in collaboration with the author [10]; we used dynamical methods.
(viii) Finally, we consider the case of an abelian scheme A A A\mathcal{A}A over a base S S SSS of arbitrary dimension. This setting was studied recently in work of AndréCorvaja-Zannier [3]. Moreover, the work of Gao on the Ax-Schanuel Theorem [30] for the universal family of abelian varieties led him to formulate a geometric condition [29] that guarantees nondegeneracy. It proves crucial in the application to Mazur's question and we will return to this point. Gao's result also relied on o-minimal geometry and the Pila-Wilkie Theorem.

4.2. Comparing the Weil and Néron-Tate heights on a subvariety

We now come to the generalization of Theorem 4.3 to nondegenerate subvarieties. We retain the notation introduced in Section 4.1. So S S SSS is a smooth irreducible quasiprojective variety defined over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯ equipped with an immersion in P m P m P^(m)\mathbb{P}^{m}Pm. We have a height h S h S h_(S)h_{S}hS on S ¯ ( Q ¯ ) S ¯ ( Q ¯ ) bar(S)( bar(Q))\bar{S}(\overline{\mathbb{Q}})S¯(Q¯). Moreover, π : A S Ï€ : A → S pi:Ararr S\pi: \mathscr{A} \rightarrow SÏ€:A→S is an abelian scheme over S S SSS presented with an immersion A P n × S A → P n × S ArarrP^(n)xx S\mathscr{A} \rightarrow \mathbb{P}^{n} \times SA→Pn×S over S S SSS. Finally, L L L\mathscr{L}L is as in Section 4 and h ^ A h ^ A hat(h)_(A)\hat{h}_{\mathscr{A}}h^A is the fiberwise Néron-Tate height on A ( Q ¯ ) A ( Q ¯ ) A( bar(Q))\mathcal{A}(\overline{\mathbb{Q}})A(Q¯).
We assume that A A A\mathscr{A}A carries symplectic level- ℓ ℓ\ellℓ structure for some fixed 3 ℓ ≥ 3 ℓ >= 3\ell \geq 3ℓ≥3 and that L L L\mathscr{L}L induces a principal polarization. For the proof of Theorem 1.12, it suffices to have the following height bound under these conditions. We also refer to [24, THEOREM B.1] for a version that relaxes some of the conditions.
Theorem 4.6 ([24, THEOREM 1.6]). Let V V VVV be a nondegenerate irreducible subvariety of A A A\mathcal{A}A that dominates S S SSS. There exist c ( V ) > 0 , c ( V ) 0 c ( V ) > 0 , c ′ ( V ) ≥ 0 c(V) > 0,c^(')(V) >= 0c(V)>0, c^{\prime}(V) \geq 0c(V)>0,c′(V)≥0, and a Zariski open and dense subset U V U ⊆ V U sube VU \subseteq VU⊆V with
h S ( π ( P ) ) c ( V ) h ^ A ( P ) + c ( V ) for all P U ( Q ¯ ) h S ( Ï€ ( P ) ) ≤ c ( V ) h ^ A ( P ) + c ′ ( V )  for all  P ∈ U ( Q ¯ ) h_(S)(pi(P)) <= c(V) hat(h)_(A)(P)+c^(')(V)quad" for all "P in U( bar(Q))h_{S}(\pi(P)) \leq c(V) \hat{h}_{\mathcal{A}}(P)+c^{\prime}(V) \quad \text { for all } P \in U(\overline{\mathbb{Q}})hS(Ï€(P))≤c(V)h^A(P)+c′(V) for all P∈U(Q¯)
We refer to Yuan and Zhang's Theorem 6.2.2 [67] for a height inequality in the dynamical setting.
Here are just a few words on the proof of Theorem 4.6. Siu's Criterion, see Remark 2.5, is used to compare the Weil height of π ( P ) Ï€ ( P ) pi(P)\pi(P)Ï€(P) with a Weil height of P P PPP. The nondegeneracy hypothesis is used to extract a volume estimate. The upshot is a lower bound for the top self-intersection number in Siu's Criterion. The predecessor of Theorem 4.6 in the earlier works [ 33 , 35 ] [ 33 , 35 ] [33,35][33,35][33,35] was proved by counting torsion points using the Geometry of Number; volumes played an important role here as well. Passing from the Weil to the Néron-Tate height introduces an additional dependency on the height of π ( P ) Ï€ ( P ) pi(P)\pi(P)Ï€(P), see (4.1). However, this contribution can be eliminated by using Masser's "ruthless strategy of killing Zimmer constants" [68, APPENDIX c]. This task is done by repeated iteration of the duplication morphism [2] which has the effect of truncating Tate's Limit Process after finitely many steps. Our ambient group scheme A A A\mathscr{A}A is quasiprojective but in general not projective. So a suitable compactification is required that admits some compatibility with the duplication morphism.
The positive constant c ( V ) c ( V ) c(V)c(V)c(V) in Theorem 4.6 ultimately comes from the application of Siu's Criterion. As such it can expressed in geometric terms.

5. APPLICATION TO MODERATE POINTS ON CURVES

In this section we sketch the main lines of the proof of Theorem 1.12. It will be enough to bound the number of moderate points, see Section 3.

5.1. The Faltings-Zhang morphism

Smooth curves of genus g 2 g ≥ 2 g >= 2g \geq 2g≥2 defined over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯ are classified by the Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯-points of a quasiprojective variety, the coarse moduli space. For us it is convenient to work with symplectic level- â„“ â„“\ellâ„“ structure on the Jacobian for some fixed integer 3 â„“ ≥ 3 â„“ >= 3\ell \geq 3ℓ≥3. With this extra data, we obtain a fine moduli space M g M g M_(g)\mathbb{M}_{g}Mg, together with a universal family C g M g C g → M g C_(g)rarrM_(g)\mathfrak{C}_{g} \rightarrow \mathbb{M}_{g}Cg→Mg. Fibers of this family are smooth curves of genus g g ggg with the said level structure on the Jacobian. Then M g M g M_(g)\mathbb{M}_{g}Mg carries the structure of a smooth quasiprojective variety of dimension 3 g 3 3 g − 3 3g-33 g-33g−3 defined over a cyclotomic field. For convenience, we replace M g M g M_(g)\mathbb{M}_{g}Mg by an irreducible component by choosing a complex root of unity of order â„“ â„“\ellâ„“ and consider it as defined over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯.
The Torelli morphism τ : M g A g Ï„ : M g → A g tau:M_(g)rarrA_(g)\tau: \mathbb{M}_{g} \rightarrow \mathbb{A}_{g}Ï„:Mg→Ag takes a smooth curve to its Jacobian with the level structure; here A g A g A_(g)\mathbb{A}_{g}Ag denotes the fine moduli space of g g ggg-dimensional abelian varieties with a principal polarization and symplectic level − â„“ -â„“-\ell−ℓ structure.
Let M 0 M ≥ 0 M >= 0M \geq 0M≥0 be an integer and consider M + 1 M + 1 M+1M+1M+1 points P 0 , , P M C g ( C ) P 0 , … , P M ∈ C g ( C ) P_(0),dots,P_(M)inC_(g)(C)P_{0}, \ldots, P_{M} \in \mathfrak{C}_{g}(\mathbb{C})P0,…,PM∈Cg(C) in the same fiber C C CCC of C g M g C g → M g C_(g)rarrM_(g)\mathfrak{C}_{g} \rightarrow \mathbb{M}_{g}Cg→Mg. The differences [ P 1 ] [ P 0 ] , , [ P M ] [ P 0 ] P 1 − P 0 , … , P M − P 0 [P_(1)]-[P_(0)],dots,[P_(M)]-[P_(0)]\left[P_{1}\right]-\left[P_{0}\right], \ldots,\left[P_{M}\right]-\left[P_{0}\right][P1]−[P0],…,[PM]−[P0] are divisors of degree 0 on C C CCC. We obtain M M MMM complex points in the Jacobian of C C CCC and so M M MMM complex points of V g V g V_(g)\mathfrak{V}_{g}Vg. We obtain a commutative diagram

of morphisms of schemes; here the exponent [ M ] [ M ] [M][M][M] denotes taking the M M MMM th fibered power over the base. The morphism D D D\mathscr{D}D is called the Faltings-Zhang morphism; see [26, LEMMA 4.1] and [71, LEMMA 3.1] for important applications to diophantine geometry of variants of this morphism. The morphism D D D\mathscr{D}D is proper.
A modified version of this construction is also useful. Say S M g S → M g S rarrM_(g)S \rightarrow \mathbb{M}_{g}S→Mg is a quasifinite morphism with S S SSS an irreducible quasiprojective variety defined over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯. We obtain a proper morphism D : C g [ M + 1 ] × M g S V g [ M ] × A g S D : C g [ M + 1 ] × M g S → V g [ M ] × A g S D:C_(g)^([M+1])xx_(M_(g))S rarrV_(g)^([M])xx_(A_(g))S\mathscr{D}: \mathfrak{C}_{g}^{[M+1]} \times_{\mathbb{M}_{g}} S \rightarrow \mathscr{V}_{g}^{[M]} \times_{\mathbb{A}_{g}} SD:Cg[M+1]×MgS→Vg[M]×AgS, again called Faltings-Zhang morphism.
Gao, using his Ax-Schanuel Theorem for the universal family N g N g N_(g)\mathfrak{N}_{g}Ng [30] and a characterization [28] of bialgebraic subvarieties of g ℜ g ℜ_(g)\mathfrak{\Re}_{g}ℜg, obtained
Theorem 5.1 (Gao [29]). Let S M g S → M g S rarrM_(g)S \rightarrow \mathbb{M}_{g}S→Mg be as above, i.e., a quasifinite morphism from an irreducible quasiprojective variety S S SSS defined over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯ and g 2 g ≥ 2 g >= 2g \geq 2g≥2. If M dim M g + 1 = M ≥ dim ⁡ M g + 1 = M >= dim M_(g)+1=M \geq \operatorname{dim} \mathbb{M}_{g}+1=M≥dim⁡Mg+1= 3 g 2 3 g − 2 3g-23 g-23g−2, then D ( E g [ M + 1 ] × M g S ) D E g [ M + 1 ] × M g S D(E_(g)^([M+1])xx_(M_(g))S)\mathscr{D}\left(\mathfrak{E}_{g}^{[M+1]} \times_{\mathbb{M}_{g}} S\right)D(Eg[M+1]×MgS) is a nondegenerate subvariety of I g [ M ] × A g S I g [ M ] × A g S I_(g)^([M])xx_(A_(g))S\mathfrak{I}_{g}^{[M]} \times_{\mathbb{A}_{g}} SIg[M]×AgS.
Mok, Pila, and Tsimerman [47] earlier proved an Ax-Schanuel Theorem for Shimura varieties. Gao's result [30] is a "mixed" version in the abelian setting. We refer to the survey [4] on recent developments in functional transcendence.
The hypothesis g 2 g ≥ 2 g >= 2g \geq 2g≥2 is crucial. The definition of the Faltings-Zhang morphism makes sense for g = 1 g = 1 g=1g=1g=1. But it will be surjective and the image is degenerate expect in the (for our purposes uninteresting) case dim S = 0 dim ⁡ S = 0 dim S=0\operatorname{dim} S=0dim⁡S=0.
We consider here for simplicity only the case S = M g S = M g S=M_(g)S=\mathbb{M}_{g}S=Mg.
Using basic dimension theory, we see dim D ( S g [ M + 1 ] ) M + 1 + dim M g dim ⁡ D S g [ M + 1 ] ≤ M + 1 + dim ⁡ M g dim D(S_(g)^([M+1])) <= M+1+dim M_(g)\operatorname{dim} \mathscr{D}\left(\mathfrak{S}_{g}^{[M+1]}\right) \leq M+1+\operatorname{dim} \mathbb{M}_{g}dim⁡D(Sg[M+1])≤M+1+dim⁡Mg. The image lies in the fibered power V g [ M ] V g [ M ] V_(g)^([M])\mathfrak{V}_{g}^{[M]}Vg[M] where the relative dimension is M g M g MgM gMg. A necessary condition for D ( C g [ M + 1 ] ) D C g [ M + 1 ] D(C_(g)^([M+1]))\mathscr{D}\left(\mathfrak{C}_{g}^{[M+1]}\right)D(Cg[M+1]) to be nondegenerate is dim D ( C g [ M + 1 ] ) M g dim ⁡ D C g [ M + 1 ] ≤ M g dim D(C_(g)^([M+1])) <= Mg\operatorname{dim} \mathscr{D}\left(\mathfrak{C}_{g}^{[M+1]}\right) \leq M gdim⁡D(Cg[M+1])≤Mg, see Example 4.5(ii). This inequality follows from
(5.1) M + 3 g 2 = M + 1 + dim M g M g (5.1) M + 3 g − 2 = M + 1 + dim ⁡ M g ≤ M g {:(5.1)M+3g-2=M+1+dim M_(g) <= Mg:}\begin{equation*} M+3 g-2=M+1+\operatorname{dim} \mathbb{M}_{g} \leq M g \tag{5.1} \end{equation*}(5.1)M+3g−2=M+1+dim⁡Mg≤Mg
If M 3 M ≤ 3 M <= 3M \leq 3M≤3, the numerical condition (5.1) is not satisfied for any g 2 g ≥ 2 g >= 2g \geq 2g≥2. For this reason, we cannot hope to work with the image of C g × M g C g C g × M g C g C_(g)xx_(M_(g))C_(g)\mathfrak{C}_{g} \times_{\mathbb{M}_{g}} \mathfrak{C}_{g}Cg×MgCg in V g V g Vg\mathfrak{V} gVg by taking differences. Moreover, there seems to be no reasonable way to work with a single copy of C g C g C_(g)\mathfrak{C}_{g}Cg, where the relations between dimensions would be even worse. The numerical condition (5.1) is satisfied for all M 4 M ≥ 4 M >= 4M \geq 4M≥4 and all g 2 g ≥ 2 g >= 2g \geq 2g≥2. Gao's Theorem implies that M 3 g 2 M ≥ 3 g − 2 M >= 3g-2M \geq 3 g-2M≥3g−2 is sufficient to guarantee nondegeneracy.
We can thus apply Theorem 4.6 to the image D ( C g [ M + 1 ] ) D C g [ M + 1 ] D(C_(g)^([M+1]))\mathscr{D}\left(\mathfrak{C}_{g}^{[M+1]}\right)D(Cg[M+1]) of the Faltings-Zhang morphism in V g [ M ] × A g M g V g [ M ] × A g M g V_(g)^([M])xx_(A_(g))M_(g)\mathscr{V}_{g}^{[M]} \times_{\mathbb{A}_{g}} \mathbb{M}_{g}Vg[M]×AgMg. Let M = 3 g 2 M = 3 g − 2 M=3g-2M=3 g-2M=3g−2, then
(5.2) h M g ( s ) c ( g ) ( h ^ N g ( P 1 P 0 ) + + h ^ O g ( P M P 0 ) ) + c ( g ) (5.2) h M g ( s ) ≤ c ( g ) h ^ N g P 1 − P 0 + ⋯ + h ^ O g P M − P 0 + c ′ ( g ) {:(5.2)h_(M_(g))(s) <= c(g)( hat(h)_(N_(g))(P_(1)-P_(0))+cdots+ hat(h)_(O_(g))(P_(M)-P_(0)))+c^(')(g):}\begin{equation*} h_{\mathbb{M}_{g}}(s) \leq c(g)\left(\hat{h}_{\mathscr{N}_{g}}\left(P_{1}-P_{0}\right)+\cdots+\hat{h}_{\mathscr{O}_{g}}\left(P_{M}-P_{0}\right)\right)+c^{\prime}(g) \tag{5.2} \end{equation*}(5.2)hMg(s)≤c(g)(h^Ng(P1−P0)+⋯+h^Og(PM−P0))+c′(g)
for all ( P 0 , , P M ) U ( Q ¯ ) P 0 , … , P M ∈ U ( Q ¯ ) (P_(0),dots,P_(M))in U( bar(Q))\left(P_{0}, \ldots, P_{M}\right) \in U(\overline{\mathbb{Q}})(P0,…,PM)∈U(Q¯) above s M g ( Q ¯ ) s ∈ M g ( Q ¯ ) s inM_(g)( bar(Q))s \in \mathbb{M}_{g}(\overline{\mathbb{Q}})s∈Mg(Q¯) where U U UUU is a Zariski open and dense subset of D ( C g [ M + 1 ] ) D C g [ M + 1 ] D(C_(g)^([M+1]))\mathscr{D}\left(\mathfrak{C}_{g}^{[M+1]}\right)D(Cg[M+1]). The constants c ( g ) > 0 c ( g ) > 0 c(g) > 0c(g)>0c(g)>0 and c ( g ) 0 c ′ ( g ) ≥ 0 c^(')(g) >= 0c^{\prime}(g) \geq 0c′(g)≥0 depend on the various choices made regarding projective immersions of M g M g M_(g)\mathbb{M}_{g}Mg and N l g N l g Nl_(g)\mathfrak{N l}_{g}Nlg. Ultimately, they depend only on g g ggg once these choices have been made.
The Zariski open U U UUU cannot be replaced by D ( C g [ M + 1 ] ) D C g [ M + 1 ] D(C_(g)^([M+1]))\mathscr{D}\left(\mathfrak{C}_{g}^{[M+1]}\right)D(Cg[M+1]). Indeed, the right-hand side of (5.2) vanishes on the diagonal P 0 = P 1 = = P M P 0 = P 1 = ⋯ = P M P_(0)=P_(1)=cdots=P_(M)P_{0}=P_{1}=\cdots=P_{M}P0=P1=⋯=PM whereas the left-hand side is unbounded as s s sss varies.
Let us shift back to using ∥ ⋅ ∥ ||*||\|\cdot\|∥⋅∥ to denote the square root of the Néron-Tate height, see Section 2.2. Let us assume that
(5.3) h M g ( s ) 2 c ( g ) (5.3) h M g ( s ) ≥ 2 c ′ ( g ) {:(5.3)h_(M_(g))(s) >= 2c^(')(g):}\begin{equation*} h_{\mathbb{M}_{g}}(s) \geq 2 c^{\prime}(g) \tag{5.3} \end{equation*}(5.3)hMg(s)≥2c′(g)
As 2 M = 6 g 4 2 M = 6 g − 4 2M=6g-42 M=6 g-42M=6g−4 we find
(5.4) h M g ( s ) c ( g ) ( 6 g 4 ) max 1 j M P j P 0 2 for all ( P 0 , , P M ) U ( Q ¯ ) (5.4) h M g ( s ) ≤ c ( g ) ( 6 g − 4 ) max 1 ≤ j ≤ M   P j − P 0 2  for all  P 0 , … , P M ∈ U ( Q ¯ ) {:(5.4)h_(M_(g))(s) <= c(g)(6g-4)max_(1 <= j <= M)||P_(j)-P_(0)||^(2)quad" for all "(P_(0),dots,P_(M))in U( bar(Q)):}\begin{equation*} h_{\mathbb{M}_{g}}(s) \leq c(g)(6 g-4) \max _{1 \leq j \leq M}\left\|P_{j}-P_{0}\right\|^{2} \quad \text { for all }\left(P_{0}, \ldots, P_{M}\right) \in U(\overline{\mathbb{Q}}) \tag{5.4} \end{equation*}(5.4)hMg(s)≤c(g)(6g−4)max1≤j≤M∥Pj−P0∥2 for all (P0,…,PM)∈U(Q¯)
Morally, (5.4) states that among a ( 3 g 1 ) 3 g − 1 ) 3g-1)3 g-1)3g−1)-tuple of points on a curve of genus g g ggg in general position, there must be a pair that repels one another with respect to the norm ∥ ⋅ ∥ ||*||\|\cdot\|∥⋅∥. The squared distance of such a pair is larger than a positive multiple, depending only on g g ggg, of the modular height h M g ( s ) h M g ( s ) h_(M_(g))(s)h_{\mathbb{M}_{g}}(s)hMg(s); this is the key to bounding the number of moderate points from Section 3.
As stated at the end of Section 4.2, the value c ( g ) c ( g ) c(g)c(g)c(g) can be expressed in terms of geometry properties of the image of S g [ M + 1 ] S g [ M + 1 ] S_(g)^([M+1])\mathfrak{S}_{g}^{[M+1]}Sg[M+1] under the Faltings-Zhang morphism.
Question 5.2. What is an admissible value for c ( g ) c ( g ) c(g)c(g)c(g) ?

5.2. Bounding the number of moderate points-a sketch

Recall that, by the discussion at the end of Section 3, we need to bound the number of moderate points.
We retain the notation of Sections 3 and 5. The curve C C CCC from Section 3 can be equipped with suitable level structure over a field F / F F ′ / F F^(')//FF^{\prime} / FF′/F with [ F : F ] F ′ : F [F^('):F]\left[F^{\prime}: F\right][F′:F] bounded in terms of g g ggg. The rank of Jac ( C ) ( F ) Jac ⁡ ( C ) F ′ Jac(C)(F^('))\operatorname{Jac}(C)\left(F^{\prime}\right)Jac⁡(C)(F′) may be dangerously larger than the rank of Jac ( C ) ( F ) Jac ⁡ ( C ) ( F ) Jac(C)(F)\operatorname{Jac}(C)(F)Jac⁡(C)(F). But recall that we are interested in bounding # C ( F ) # C ( F ) #C(F)\# C(F)#C(F) from above, so only the group Jac ( C ) ( F ) group ⁡ Jac ⁡ ( C ) ( F ) group Jac(C)(F)\operatorname{group} \operatorname{Jac}(C)(F)group⁡Jac⁡(C)(F) will be relevant. Moreover, c 1 , c 2 c 1 , c 2 c_(1),c_(2)c_{1}, c_{2}c1,c2, and c 3 c 3 c_(3)c_{3}c3 from a suitable version of Vojta's inequality are unaffected by extending F F FFF. The effect is that we may identify C C CCC with a fiber of C g C g C_(g)\mathfrak{C}_{g}Cg above some point s M g ( F ) s ∈ M g F ′ s inM_(g)(F^('))s \in \mathbb{M}_{g}\left(F^{\prime}\right)s∈Mg(F′). For simplicity, we assume F = F F = F ′ F=F^(')F=F^{\prime}F=F′ for this proof sketch.
We require some additional information on c 3 ( C ) c 3 ( C ) c_(3)(C)c_{3}(C)c3(C). It turns out that we can take c 3 = c 4 ( g ) max { 1 , h M g ( s ) } c 3 = c 4 ( g ) max 1 , h M g ( s ) c_(3)=c_(4)(g)max{1,h_(M_(g))(s)}c_{3}=c_{4}(g) \max \left\{1, h_{\mathbb{M}_{g}}(s)\right\}c3=c4(g)max{1,hMg(s)} where c 4 ( g ) > 0 c 4 ( g ) > 0 c_(4)(g) > 0c_{4}(g)>0c4(g)>0 depends on g g ggg. This follows Rémond's work [55] on the Vojta inequality. A similar dependency is apparent in de Diego's result, Theorem 1.6.
Suppose now that (5.3) holds, so h M g ( s ) h M g ( s ) h_(M_(g))(s)h_{\mathbb{M}_{g}}(s)hMg(s) is sufficiently large in terms of g g ggg. We fix an auxiliary base point P C ( F ) P ′ ∈ C ( F ) P^(')in C(F)P^{\prime} \in C(F)P′∈C(F). We must bound from above the number of points in
B ( R ) = { P C ( F ) : P P 2 R 2 } with R = ( c 4 ( g ) h M g ( s ) ) 1 / 2 B ( R ) = P ∈ C ( F ) : P − P ′ 2 ≤ R 2  with  R = c 4 ( g ) h M g ( s ) 1 / 2 B(R)={P in C(F):||P-P^(')||^(2) <= R^(2)}quad" with "R=(c_(4)(g)h_(M_(g))(s))^(1//2)B(R)=\left\{P \in C(F):\left\|P-P^{\prime}\right\|^{2} \leq R^{2}\right\} \quad \text { with } R=\left(c_{4}(g) h_{\mathbb{M}_{g}}(s)\right)^{1 / 2}B(R)={P∈C(F):∥P−P′∥2≤R2} with R=(c4(g)hMg(s))1/2
where ∥ ⋅ ∥ ||*||\|\cdot\|∥⋅∥ denotes the square root of the Néron-Tate height on the fiber of N g A g N g → A g N_(g)rarrA_(g)\mathscr{N}_{g} \rightarrow \mathbb{A}_{g}Ng→Ag associated to the Jacobian of C C CCC.
Recall M = 3 g 2 M = 3 g − 2 M=3g-2M=3 g-2M=3g−2 and suppose P 0 , , P M C ( F ) P 0 , … , P M ∈ C ( F ) P_(0),dots,P_(M)in C(F)P_{0}, \ldots, P_{M} \in C(F)P0,…,PM∈C(F). If the tuple ( P 0 , , P M ) P 0 , … , P M (P_(0),dots,P_(M))\left(P_{0}, \ldots, P_{M}\right)(P0,…,PM) is in general position, i.e., ( P 1 P 0 , , P M P 0 ) P 1 − P 0 , … , P M − P 0 (P_(1)-P_(0),dots,P_(M)-P_(0))\left(P_{1}-P_{0}, \ldots, P_{M}-P_{0}\right)(P1−P0,…,PM−P0) lies in U ( Q ¯ ) U ( Q ¯ ) U( bar(Q))U(\overline{\mathbb{Q}})U(Q¯) from (5.4), then there is i i iii with
P i B ( P 0 , r ) = { P C ( F ) : P P 0 2 r 2 } with r = ( c 5 ( g ) h M g ( s ) ) 1 / 2 P i ∉ B P 0 , r = P ∈ C ( F ) : P − P 0 2 ≤ r 2  with  r = c 5 ( g ) h M g ( s ) 1 / 2 P_(i)!in B(P_(0),r)={P in C(F):||P-P_(0)||^(2) <= r^(2)}quad" with "r=(c_(5)(g)h_(M_(g))(s))^(1//2)P_{i} \notin B\left(P_{0}, r\right)=\left\{P \in C(F):\left\|P-P_{0}\right\|^{2} \leq r^{2}\right\} \quad \text { with } r=\left(c_{5}(g) h_{\mathbb{M}_{g}}(s)\right)^{1 / 2}Pi∉B(P0,r)={P∈C(F):∥P−P0∥2≤r2} with r=(c5(g)hMg(s))1/2
If we had a guarantee that such ( M + 1 ) ( M + 1 ) (M+1)(M+1)(M+1)-tuples of pairwise distinct points are always in general position, then # B ( P 0 , r ) < M = 3 g 2 # B P 0 , r < M = 3 g − 2 #B(P_(0),r) < M=3g-2\# B\left(P_{0}, r\right)<M=3 g-2#B(P0,r)<M=3g−2. By sphere packing, we can cover the image of B ( R ) B ( R ) B(R)B(R)B(R) in Jac ( C ) ( F ) R Jac ⁡ ( C ) ( F ) ⊗ R Jac(C)(F)oxR\operatorname{Jac}(C)(F) \otimes \mathbb{R}Jac⁡(C)(F)⊗R by at most ( 1 + 2 R / r ) r k J a c ( C ) ( F ) ( 1 + 2 R / r ) r k J a c ( C ) ( F ) (1+2R//r)^(rkJac(C)(F))(1+2 R / r)^{\mathrm{rkJac}(C)(F)}(1+2R/r)rkJac(C)(F) closed balls in Jac ( C ) ( F ) Jac ⁡ ( C ) ( F ) Jac(C)(F)\operatorname{Jac}(C)(F)Jac⁡(C)(F) of radius r r rrr. One can even arrange for the ball centers to arise as points of C ( F ) C ( F ) C(F)C(F)C(F). The modular height h M g ( s ) h M g ( s ) h_(M_(g))(s)h_{\mathbb{M}_{g}}(s)hMg(s) cancels out in the quotient
R r = ( c 4 ( g ) c 5 ( g ) ) 1 / 2 R r = c 4 ( g ) c 5 ( g ) 1 / 2 (R)/(r)=((c_(4)(g))/(c_(5)(g)))^(1//2)\frac{R}{r}=\left(\frac{c_{4}(g)}{c_{5}(g)}\right)^{1 / 2}Rr=(c4(g)c5(g))1/2
This would complete the proof of Theorem 1.12 except that there is no reason to believe that ( P 1 P 0 , , P M P 0 ) U ( Q ¯ ) P 1 − P 0 , … , P M − P 0 ∈ U ( Q ¯ ) (P_(1)-P_(0),dots,P_(M)-P_(0))in U( bar(Q))\left(P_{1}-P_{0}, \ldots, P_{M}-P_{0}\right) \in U(\overline{\mathbb{Q}})(P1−P0,…,PM−P0)∈U(Q¯) (even if the P j P j P_(j)P_{j}Pj are pairwise distinct). Treating points with image in the complement of U U UUU requires induction on the dimension. Here we rely on the freedom to replace M g M g M_(g)\mathbb{M}_{g}Mg by a subvariety in that Gao's Theorem 5.1.
Let us briefly explain the resulting induction step. Observe that the dimension of this exceptional set is at most dim D ( C g [ M + 1 ] ) 1 M + dim M g dim ⁡ D C g [ M + 1 ] − 1 ≤ M + dim ⁡ M g dim D(C_(g)^([M+1]))-1 <= M+dim M_(g)\operatorname{dim} \mathscr{D}\left(\mathfrak{C}_{g}^{[M+1]}\right)-1 \leq M+\operatorname{dim} \mathbb{M}_{g}dim⁡D(Cg[M+1])−1≤M+dim⁡Mg. There are two cases for ( P 0 , , P M ) P 0 , … , P M (P_(0),dots,P_(M))\left(P_{0}, \ldots, P_{M}\right)(P0,…,PM) with image in the exceptional set ( D ( C g [ M + 1 ] ) U ) ( Q ¯ ) D C g [ M + 1 ] ∖ U ( Q ¯ ) (D(C_(g)^([M+1]))\\U)( bar(Q))\left(\mathscr{D}\left(\mathfrak{C}_{g}^{[M+1]}\right) \backslash U\right)(\overline{\mathbb{Q}})(D(Cg[M+1])∖U)(Q¯) on which we do not have the height inequality. For the case study, recall that s M g ( Q ¯ ) s ∈ M g ( Q ¯ ) s inM_(g)( bar(Q))s \in \mathbb{M}_{g}(\overline{\mathbb{Q}})s∈Mg(Q¯) denotes the point below all the P j P j P_(j)P_{j}Pj and τ ( s ) A g ( Q ¯ ) Ï„ ( s ) ∈ A g ( Q ¯ ) tau(s)inA_(g)( bar(Q))\tau(s) \in \mathbb{A}_{g}(\overline{\mathbb{Q}})Ï„(s)∈Ag(Q¯) is its image under the Torelli morphism τ Ï„ tau\tauÏ„.
First, assume that the fiber of D ( C g [ M + 1 ] ) U A g D C g [ M + 1 ] ∖ U → A g D(C_(g)^([M+1]))\\U rarrAg\mathscr{D}\left(\mathfrak{C}_{g}^{[M+1]}\right) \backslash U \rightarrow \mathbb{A} gD(Cg[M+1])∖U→Ag above τ ( s ) Ï„ ( s ) tau(s)\tau(s)Ï„(s) has dimension at most M M MMM. This fiber contains ( P 1 P 0 , , P M P 0 ) P 1 − P 0 , … , P M − P 0 (P_(1)-P_(0),dots,P_(M)-P_(0))\left(P_{1}-P_{0}, \ldots, P_{M}-P_{0}\right)(P1−P0,…,PM−P0). This case is solved using a zero estimate motivated by the following simple lemma.
Lemma 5.3. Suppose C C CCC is an irreducible curve defined over C C C\mathbb{C}C and W W WWW a proper Zariski closed subset of C M C M C^(M)C^{M}CM. If Σ C ( C ) Σ ⊆ C ( C ) Sigma sube C(C)\Sigma \subseteq C(\mathbb{C})Σ⊆C(C) with Σ M W ( C ) Σ M ⊆ W ( C ) Sigma^(M)sube W(C)\Sigma^{M} \subseteq W(\mathbb{C})ΣM⊆W(C), then Σ Î£ Sigma\SigmaΣ is finite.
This statement can be quantified if C C CCC is presented as a curve in some projective space. Using Bézout's Theorem, one can show that # Σ # Σ #Sigma\# \Sigma#Σ is bounded from above in terms of the degrees of C C CCC and W W WWW. In our application, both degrees will be uniformly bounded as all varieties arise in algebraic families. This ultimately leads to the desired uniformity estimates.
The second case is if the fiber of D ( C g [ M + 1 ] ) U A g D C g [ M + 1 ] ∖ U → A g D(C_(g)^([M+1]))\\U rarrA_(g)\mathscr{D}\left(\mathfrak{C}_{g}^{[M+1]}\right) \backslash U \rightarrow \mathbb{A}_{g}D(Cg[M+1])∖U→Ag above τ ( s ) Ï„ ( s ) tau(s)\tau(s)Ï„(s) has dimension at least M + 1 M + 1 M+1M+1M+1. For dimension reasons, s s sss lies in a proper subvariety S S SSS of M g M g M_(g)\mathbb{M}_{g}Mg. Here we apply induction on the dimension and replace M g M g M_(g)\mathbb{M}_{g}Mg by its subvariety S S SSS.
This completes the proof sketch.
Kühne [39] combined ideas from equidistribution with the approach laid out in [24] to get a suitable uniform estimate for # B ( P 0 , r ) # B P 0 , r #B(P_(0),r)\# B\left(P_{0}, r\right)#B(P0,r) without the restriction (5.3) on h M g ( s ) h M g ( s ) h_(M_(g))(s)h_{\mathbb{M}_{g}}(s)hMg(s). Yuan's Theorem 1.1 [66] does so as well, but he follows a different approach. He obtains a more general estimate that works also in the function field setting and allows for a larger R R RRR.

6. HYPERELLIPTIC CURVES

A hyperelliptic curve is a smooth curve of genus at least 2 that admits a degree 2 morphism to the projective line. Hyperelliptic curves have particularly simple planar models. Indeed, if the base field is a number field F F FFF, then a hyperelliptic curve of genus g g ggg can be
represented by a hyperelliptic equation
Y 2 = f ( X ) Y 2 = f ( X ) Y^(2)=f(X)quadY^{2}=f(X) \quadY2=f(X) with f F [ X ] f ∈ F [ X ] f in F[X]f \in F[X]f∈F[X] monic and square-free of degree 2 g + 1 2 g + 1 2g+12 g+12g+1 or 2 g + 2 2 g + 2 2g+22 g+22g+2.
In this section we determine consequences of Theorem 1.12 for hyperelliptic curves. Our aim is to leave the world of curves and Jacobians and to present a bound for the number of rational solutions of Y 2 = f ( X ) Y 2 = f ( X ) Y^(2)=f(X)Y^{2}=f(X)Y2=f(X) that can be expressed in terms of f f fff. We refer to Section 6 of [23] for a similar example in a 1-parameter family of hyperelliptic curves.
To keep technicalities to a minimum, we assume that our base field is F = Q F = Q F=QF=\mathbb{Q}F=Q and that f Z [ X ] f ∈ Z [ X ] f inZ[X]f \in \mathbb{Z}[X]f∈Z[X] is monic of degree d = 2 g + 1 d = 2 g + 1 d=2g+1d=2 g+1d=2g+1 and factors into linear factors in Q [ X ] Q [ X ] Q[X]\mathbb{Q}[X]Q[X]. The curve represented by the hyperelliptic equation has a marked Weierstrass point "at infinity." These assumptions can be loosened with some extra effort. For example, if f f fff does not factor in Q [ X ] Q [ X ] Q[X]\mathbb{Q}[X]Q[X], then the class number of the splitting field will play a part.
Say, f = X d + f d 1 X d 1 + + f 0 f = X d + f d − 1 X d − 1 + ⋯ + f 0 f=X^(d)+f_(d-1)X^(d-1)+cdots+f_(0)f=X^{d}+f_{d-1} X^{d-1}+\cdots+f_{0}f=Xd+fd−1Xd−1+⋯+f0. By the assumption above, f = ( X α 1 ) ( X α d ) f = X − α 1 ⋯ X − α d f=(X-alpha_(1))cdots(X-alpha_(d))f=\left(X-\alpha_{1}\right) \cdots\left(X-\alpha_{d}\right)f=(X−α1)⋯(X−αd) with α 1 , , α d Q α 1 , … , α d ∈ Q alpha_(1),dots,alpha_(d)inQ\alpha_{1}, \ldots, \alpha_{d} \in \mathbb{Q}α1,…,αd∈Q which are necessarily integers. The discriminant of f f fff is
Δ f = 1 i < j d ( α j α i ) 2 Z { 0 } Δ f = ∏ 1 ≤ i < j ≤ d   α j − α i 2 ∈ Z ∖ { 0 } Delta_(f)=prod_(1 <= i < j <= d)(alpha_(j)-alpha_(i))^(2)inZ\\{0}\Delta_{f}=\prod_{1 \leq i<j \leq d}\left(\alpha_{j}-\alpha_{i}\right)^{2} \in \mathbb{Z} \backslash\{0\}Δf=∏1≤i<j≤d(αj−αi)2∈Z∖{0}
The Mordell Conjecture applied to the hyperelliptic curve represented by Y 2 = f ( X ) Y 2 = f ( X ) Y^(2)=f(X)Y^{2}=f(X)Y2=f(X) states
# { ( x , y ) Q 2 : y 2 = f ( x ) } < # ( x , y ) ∈ Q 2 : y 2 = f ( x ) < ∞ #{(x,y)inQ^(2):y^(2)=f(x)} < oo\#\left\{(x, y) \in \mathbb{Q}^{2}: y^{2}=f(x)\right\}<\infty#{(x,y)∈Q2:y2=f(x)}<∞
We have the following estimate for the cardinality. Below ω ( n ) ω ( n ) omega(n)\omega(n)ω(n) denotes the number of distinct prime divisors of n Z { 0 } n ∈ Z ∖ { 0 } n inZ\\{0}n \in \mathbb{Z} \backslash\{0\}n∈Z∖{0}.
Theorem 6.1. Let g 2 g ≥ 2 g >= 2g \geq 2g≥2. There exist c ( g ) > 1 c ( g ) > 1 c(g) > 1c(g)>1c(g)>1 and c ( g ) > 0 c ′ ( g ) > 0 c^(')(g) > 0c^{\prime}(g)>0c′(g)>0 with the following property. Suppose f Z [ X ] f ∈ Z [ X ] f inZ[X]f \in \mathbb{Z}[X]f∈Z[X] is monic of degree 2 g + 1 2 g + 1 2g+12 g+12g+1, square-free, and factors into linear factors in Q [ X ] Q [ X ] Q[X]\mathbb{Q}[X]Q[X]. Then
(6.1) # { ( x , y ) Q 2 : y 2 = f ( x ) } c ( g ) c ( g ) ω ( Δ f ) (6.1) # ( x , y ) ∈ Q 2 : y 2 = f ( x ) ≤ c ′ ( g ) c ( g ) ω Δ f {:(6.1)#{(x,y)inQ^(2):y^(2)=f(x)} <= c^(')(g)c(g)^(omega(Delta_(f))):}\begin{equation*} \#\left\{(x, y) \in \mathbb{Q}^{2}: y^{2}=f(x)\right\} \leq c^{\prime}(g) c(g)^{\omega\left(\Delta_{f}\right)} \tag{6.1} \end{equation*}(6.1)#{(x,y)∈Q2:y2=f(x)}≤c′(g)c(g)ω(Δf)
Proof. The hyperelliptic equation represents a curve C C CCC defined over Q Q Q\mathbb{Q}Q of genus g g ggg.
If p p ppp is a prime number with p Δ f p ∤ Δ f p∤Delta_(f)p \nmid \Delta_{f}p∤Δf, then the α i α i alpha_(i)\alpha_{i}αi are pairwise distinct modulo p p ppp. If p p ppp is also odd, then the equation Y 2 = f ( X ) Y 2 = f ( X ) Y^(2)=f(X)Y^{2}=f(X)Y2=f(X) reduced modulo p p ppp defines a hyperelliptic curve over F p F p F_(p)\mathbb{F}_{p}Fp. So C C CCC has good reduction at all primes that do not divide 2 Δ f 2 Δ f 2Delta_(f)2 \Delta_{f}2Δf. Thus the Jacobian Jac ( C ) Jac ⁡ ( C ) Jac(C)\operatorname{Jac}(C)Jac⁡(C) has good reduction at the same primes.
We may embed C C CCC into its Jacobian Jac ( C ) Jac ⁡ ( C ) Jac(C)\operatorname{Jac}(C)Jac⁡(C) by sending the marked Weierstrass point to 0 . Each root α i α i alpha_(i)\alpha_{i}αi of f f fff corresponds to a rational point in C ( Q ) C ( Q ) C(Q)C(\mathbb{Q})C(Q) and it is sent to a point of order 2 in Jac ( C ) Jac ⁡ ( C ) Jac(C)\operatorname{Jac}(C)Jac⁡(C). Moreover, these points generate the 2-torsion in Jac ( C ) tors Jac ⁡ ( C ) tors  Jac(C)_("tors ")\operatorname{Jac}(C)_{\text {tors }}Jac⁡(C)tors . In particular, all points of order 2 in Jac ( C ) tors Jac ⁡ ( C ) tors  Jac(C)_("tors ")\operatorname{Jac}(C)_{\text {tors }}Jac⁡(C)tors  are rational.
Next we bound the rank of Jac ( C ) ( Q ) Jac ⁡ ( C ) ( Q ) Jac(C)(Q)\operatorname{Jac}(C)(\mathbb{Q})Jac⁡(C)(Q) from above. Indeed, we could use the work of Ooe-Top [49] or [37, THEOREM c.1.9]. The latter applied to Jac ( C ) , k = Q , m = 2 Jac ⁡ ( C ) , k = Q , m = 2 Jac(C),k=Q,m=2\operatorname{Jac}(C), k=\mathbb{Q}, m=2Jac⁡(C),k=Q,m=2, and S S SSS the prime divisors of 2 Δ f 2 Δ f 2Delta_(f)2 \Delta_{f}2Δf yields rk Jac ( C ) ( Q ) 2 g # S 2 g ω ( 2 Δ f ) 2 g + 2 g ω ( Δ f ) rk ⁡ Jac ⁡ ( C ) ( Q ) ≤ 2 g # S ≤ 2 g ω 2 Δ f ≤ 2 g + 2 g ω Δ f rk Jac(C)(Q) <= 2g#S <= 2g omega(2Delta_(f)) <= 2g+2g omega(Delta_(f))\operatorname{rk} \operatorname{Jac}(C)(\mathbb{Q}) \leq 2 g \# S \leq 2 g \omega\left(2 \Delta_{f}\right) \leq 2 g+2 g \omega\left(\Delta_{f}\right)rk⁡Jac⁡(C)(Q)≤2g#S≤2gω(2Δf)≤2g+2gω(Δf) Here we use that Q Q Q\mathbb{Q}Q has a trivial class group; in a more general setup, the class group of the
splitting field of f f fff will enter at this point. The estimate (6.1) follows from Theorem 1.12 in the case F = Q F = Q F=QF=\mathbb{Q}F=Q with adjusted constants.
It is tempting to average (6.1) over the f f fff bounded in a suitable way, e.g., by bounding the maximal modulus of the roots by a parameter X X XXX. As pointed out to the author by Christian Elsholtz and Martin Widmer, this average will be unbounded as X X → ∞ X rarr ooX \rightarrow \inftyX→∞.

ACKNOWLEDGMENTS

Much of the work presented here was obtained together with my coauthors. In particular, I thank Serge Cantat, Vesselin Dimitrov, Ziyang Gao, and Junyi Xie for the fruitful collaboration. I also thank Gabriel Dill, Ziyang Gao, Lars Kühne, and David Masser for comments on this survey.

FUNDING

The author has received funding from the Swiss National Science Foundation project nํ 200020_184623.

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PHILIPP HABEGGER

Department of Mathematics and Computer Science, University of Basel, Spiegelgasse 1, 4051 Basel, Switzerland, philipp.habegger@unibas.ch

THETA LIFTING AND LANGLANDS FUNCTORIALITY

ATSUSHI ICHINO

ABSTRACT

We review various aspects of theta lifting and its role in studying Langlands functoriality. In particular, we discuss realizations of the Jacquet-Langlands correspondence and the Shimura-Waldspurger correspondence in terms of theta lifting and their arithmetic applications.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 11F70; Secondary 11F27, 11F67, 22E50

KEYWORDS

Automorphic representations, Langlands functoriality, theta lifting

1. INTRODUCTION

Langlands functoriality is a principle relating two different kinds of automorphic forms and plays a pivotal role in number theory. Before Langlands formulated this principle in [42], this phenomenon was already observed in the following classical example discovered by Eichler [15] and developed by Shimizu [48]. Consider the space
S k ( Γ 0 ( N ) ) S k Γ 0 ( N ) S_(k)(Gamma_(0)(N))S_{k}\left(\Gamma_{0}(N)\right)Sk(Γ0(N))
of elliptic cusp forms of weight k k kkk and level N N NNN, where k k kkk and N N NNN are positive integers and Γ 0 ( N ) Γ 0 ( N ) Gamma_(0)(N)\Gamma_{0}(N)Γ0(N) is the congruence subgroup given by
Γ 0 ( N ) = { ( a b c d ) S L 2 ( Z ) | c 0 mod N } Γ 0 ( N ) = a b c d ∈ S L 2 ( Z ) c ≡ 0 mod N Gamma_(0)(N)={([a,b],[c,d])inSL_(2)(Z)|c-=0mod N}\Gamma_{0}(N)=\left\{\left.\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \in \mathrm{SL}_{2}(\mathbb{Z}) \right\rvert\, c \equiv 0 \bmod N\right\}Γ0(N)={(abcd)∈SL2(Z)|c≡0modN}
This space consists of holomorphic functions f f fff on the upper half-plane S S S\mathfrak{S}S which satisfy
f ( γ z ) = j ( γ , z ) k f ( z ) f ( γ z ) = j ( γ , z ) k f ( z ) f(gamma z)=j(gamma,z)^(k)f(z)f(\gamma z)=j(\gamma, z)^{k} f(z)f(γz)=j(γ,z)kf(z)
for all γ Γ 0 ( N ) γ ∈ Γ 0 ( N ) gamma inGamma_(0)(N)\gamma \in \Gamma_{0}(N)γ∈Γ0(N) and z S z ∈ S z inSz \in \mathfrak{S}z∈S and which vanish at all cusps. Here S L 2 ( R ) S L 2 ( R ) SL_(2)(R)\mathrm{SL}_{2}(\mathbb{R})SL2(R) acts on S S S\mathscr{S}S by linear fractional transformations and j ( ( a b c d ) , z ) = c z + d j a b c d , z = c z + d j(([a,b],[c,d]),z)=cz+dj\left(\left(\begin{array}{ll}a & b \\ c & d\end{array}\right), z\right)=c z+dj((abcd),z)=cz+d is the factor of automorphy. It is also equipped with the action of Hecke operators T n T n T_(n)T_{n}Tn for all positive integers n n nnn, which is a central tool in the arithmetic study of automorphic forms. On the other hand, to every indefinite quaternion division algebra B B BBB over Q Q Q\mathbb{Q}Q, we may associate a space
S k ( Γ B ) S k Γ B S_(k)(Gamma_(B))S_{k}\left(\Gamma_{B}\right)Sk(ΓB)
of modular forms, where Γ B Γ B Gamma_(B)\Gamma_{B}ΓB is the group of norm-one elements in B B BBB. Namely, this space is defined similarly by replacing Γ 0 ( N ) Γ 0 ( N ) Gamma_(0)(N)\Gamma_{0}(N)Γ0(N) by Γ B Γ B Gamma_(B)\Gamma_{B}ΓB (which can be regarded as a subgroup of ( B Q R ) × G L 2 ( R ) ) B ⊗ Q R × ≅ G L 2 ( R ) {:(Box_(Q)R)^(xx)~=GL_(2)(R))\left.\left(B \otimes_{\mathbb{Q}} \mathbb{R}\right)^{\times} \cong \mathrm{GL}_{2}(\mathbb{R})\right)(B⊗QR)×≅GL2(R)) and is equipped with the action of Hecke operators T n B T n B T_(n)^(B)T_{n}^{B}TnB. Now assume that N N NNN is the product of an even number of distinct primes and B B BBB is ramified precisely at the primes dividing N N NNN. Then by the works of Eichler and Shimizu, the trace of T n B T n B T_(n)^(B)T_{n}^{B}TnB on S k ( Γ B ) S k Γ B S_(k)(Gamma_(B))S_{k}\left(\Gamma_{B}\right)Sk(ΓB) coincides with the trace of T n T n T_(n)T_{n}Tn on the new part of S k ( Γ 0 ( N ) ) S k Γ 0 ( N ) S_(k)(Gamma_(0)(N))S_{k}\left(\Gamma_{0}(N)\right)Sk(Γ0(N)) for all n n nnn prime to N N NNN.
This remarkable relation was thoroughly studied by Jacquet-Langlands [37] in the framework of automorphic representations. Let F F FFF be a number field with adèle ring A A A\mathbb{A}A. Let B B BBB be a quaternion division algebra over F F FFF. Then Jacquet-Langlands proved that for any irreducible automorphic representation π B v π v B Ï€ B ≅ ⊗ v Ï€ v B pi^(B)~=ox_(v)pi_(v)^(B)\pi^{B} \cong \otimes_{v} \pi_{v}^{B}Ï€B≅⊗vÏ€vB of B × ( A ) B × ( A ) B^(xx)(A)B^{\times}(\mathbb{A})B×(A), there exists a unique irreducible automorphic representation π v π v Ï€ ≅ ⊗ v Ï€ v pi~=ox_(v)pi_(v)\pi \cong \otimes_{v} \pi_{v}π≅⊗vÏ€v of G L 2 ( A ) G L 2 ( A ) GL_(2)(A)\mathrm{GL}_{2}(\mathbb{A})GL2(A) such that
π v π v B Ï€ v ≅ Ï€ v B pi_(v)~=pi_(v)^(B)\pi_{v} \cong \pi_{v}^{B}Ï€v≅πvB
for almost all places v v vvv of F F FFF. Moreover, they described the image of this map π B π Ï€ B ↦ Ï€ pi^(B)|->pi\pi^{B} \mapsto \piÏ€B↦π precisely.
The Jacquet-Langlands correspondence gives a basic example of Langlands functoriality. To explain this, let G G GGG be a connected reductive group over F F FFF. Let L G L G ^(L)G{ }^{L} GLG be the L L LLL-group of G G GGG, which was introduced by Langlands and which should govern automorphic representations of G ( A ) G ( A ) G(A)G(\mathbb{A})G(A). Explicitly, L G L G ^(L)G{ }^{L} GLG is defined as a semiproduct G ^ Γ F G ^ ⋊ Γ F hat(G)><|Gamma_(F)\hat{G} \rtimes \Gamma_{F}G^⋊ΓF, where G ^ G ^ hat(G)\hat{G}G^ is the complex dual group of G , Γ F = Gal ( F ¯ / F ) G , Γ F = Gal ⁡ ( F ¯ / F ) G,Gamma_(F)=Gal( bar(F)//F)G, \Gamma_{F}=\operatorname{Gal}(\bar{F} / F)G,ΓF=Gal⁡(F¯/F) is the absolute Galois group of F F FFF, and the action of
Γ F Γ F Gamma_(F)\Gamma_{F}ΓF on G ^ G ^ hat(G)\hat{G}G^ is inherited from the action of Γ F Γ F Gamma_(F)\Gamma_{F}ΓF on the root datum of G G GGG. To motivate it, let us admit for the moment the existence of the hypothetical Langlands group L F L F L_(F)\mathscr{L}_{F}LF over F F FFF, which is equipped with a surjection L F Γ F L F → Γ F L_(F)rarrGamma_(F)\mathscr{L}_{F} \rightarrow \Gamma_{F}LF→ΓF. Then it is conjectured that irreducible automorphic representations of G ( A ) G ( A ) G(A)G(\mathbb{A})G(A) are classified in terms of certain L L LLL-homomorphisms L F L G L F → L G L_(F)rarr^(L)G\mathscr{L}_{F} \rightarrow{ }^{L} GLF→LG, i.e., homomorphisms commuting with the projections to Γ F Γ F Gamma_(F)\Gamma_{F}ΓF. (Strictly speaking, we consider here packets of tempered automorphic representations.) Now suppose that we have another connected reductive quasisplit group G G ′ G^(')G^{\prime}G′ over F F FFF and an L L LLL-homomorphism
r : L G L G r : L G → L G ′ r:^(L)G rarr^(L)G^(')r:{ }^{L} G \rightarrow{ }^{L} G^{\prime}r:LG→LG′
Let π Ï€ pi\piÏ€ be an irreducible automorphic representation of G ( A ) G ( A ) G(A)G(\mathbb{A})G(A) which should correspond to an L L LLL-homomorphism
ϕ : L F L G Ï• : L F → L G phi:L_(F)rarr^(L)G\phi: \mathscr{L}_{F} \rightarrow{ }^{L} GÏ•:LF→LG
Then Langlands functoriality predicts the existence of an irreducible automorphic representation π Ï€ ′ pi^(')\pi^{\prime}π′ of G ( A ) G ′ ( A ) G^(')(A)G^{\prime}(\mathbb{A})G′(A) which should correspond to the L L LLL-homomorphism
r ϕ : L F L G r ∘ Ï• : L F → L G ′ r@phi:L_(F)rarr^(L)G^(')r \circ \phi: \mathscr{L}_{F} \rightarrow{ }^{L} G^{\prime}r∘ϕ:LF→LG′
This conjectural relation between π Ï€ pi\piÏ€ and π Ï€ ′ pi^(')\pi^{\prime}π′ can be formulated without assuming the existence of L F L F L_(F)\mathscr{L}_{F}LF as follows. Recall that for almost all places v v vvv of F F FFF, the local component π v Ï€ v pi_(v)\pi_{v}Ï€v of π Ï€ pi\piÏ€ at v v vvv is unramified, so that it determines and is determined by a G ^ G ^ hat(G)\hat{G}G^-conjugacy class c ( π v ) c Ï€ v c(pi_(v))c\left(\pi_{v}\right)c(Ï€v) in L G L G ^(L)G{ }^{L} GLG via the Satake isomorphism. Then π Ï€ ′ pi^(')\pi^{\prime}π′ should satisfy
c ( π v ) = r ( c ( π v ) ) c Ï€ v ′ = r c Ï€ v c(pi_(v)^('))=r(c(pi_(v)))c\left(\pi_{v}^{\prime}\right)=r\left(c\left(\pi_{v}\right)\right)c(Ï€v′)=r(c(Ï€v))
for almost all v v vvv. Note that the Jacquet-Langlands correspondence mentioned above is the special case when G = B × , G = G L 2 G = B × , G ′ = G L 2 G=B^(xx),G^(')=GL_(2)G=B^{\times}, G^{\prime}=\mathrm{GL}_{2}G=B×,G′=GL2 (so that L G = L G = G L 2 ( C ) × Γ F L G = L G ′ = G L 2 ( C ) × Γ F ^(L)G=^(L)G^(')=GL_(2)(C)xxGamma_(F){ }^{L} G={ }^{L} G^{\prime}=\mathrm{GL}_{2}(\mathbb{C}) \times \Gamma_{F}LG=LG′=GL2(C)×ΓF ), and r r rrr is the identity map.
Although Langlands functoriality is out of reach in general, it led to substantial developments in the theory of automorphic forms. For example, the trace formula was developed by Arthur to study automorphic representations, culminating in his book [1] which establishes the case when G G GGG is a symplectic group or a quasisplit special orthogonal group, G G ′ G^(')G^{\prime}G′ is a general linear group, and r r rrr is the standard embedding. There are also other methods to attack Langlands functoriality, such as the converse theorem [12,13], the automorphic descent [24], and the theta lifting. In this report, we will discuss various aspects of the theta lifting, which can be viewed as an explicit realization in the case when ( G , G ) G , G ′ (G,G^('))\left(G, G^{\prime}\right)(G,G′) is a certain pair of classical groups.

2. THETA LIFTING

In this section, we recall the notion of the theta lifting, with emphasis on the realization of the Jacquet-Langlands correspondence. We also review some of its applications to explicit formulas for automorphic periods in terms of special values of L L LLL-functions.

2.1. Basic definitions and properties

Let F F FFF be a number field with adèle ring A = A F A = A F A=A_(F)\mathbb{A}=\mathbb{A}_{F}A=AF. Let W W WWW be a symplectic space over F F FFF equipped with a nondegenerate bilinear alternating form ( , ) W ( ⋅ , ⋅ ) W (*,*)_(W)(\cdot, \cdot)_{W}(⋅,⋅)W and let Sp ( W ) Sp ⁡ ( W ) Sp(W)\operatorname{Sp}(W)Sp⁡(W) denote the symplectic group of W W WWW. Similarly, let V V VVV be a quadratic space over F F FFF equipped with a nondegenerate bilinear symmetric form ( , ) V ( ⋅ , ⋅ ) V (*,*)_(V)(\cdot, \cdot)_{V}(⋅,⋅)V and let O ( V ) O ( V ) O(V)\mathrm{O}(V)O(V) denote the orthogonal group of V V VVV. Then the pair
( Sp ( W ) , O ( V ) ) ( Sp ⁡ ( W ) , O ( V ) ) (Sp(W),O(V))(\operatorname{Sp}(W), \mathrm{O}(V))(Sp⁡(W),O(V))
is an example of a reductive dual pair introduced by Howe [30]. Namely, if we consider the symplectic space W = W F V W = W ⊗ F V W=Wox_(F)V\mathbb{W}=W \otimes_{F} VW=W⊗FV equipped with the form ( , ) W ( , ) V ( ⋅ , ⋅ ) W ⊗ ( ⋅ , ⋅ ) V (*,*)_(W)ox(*,*)_(V)(\cdot, \cdot)_{W} \otimes(\cdot, \cdot)_{V}(⋅,⋅)W⊗(⋅,⋅)V and the natural homomorphism
Sp ( W ) × O ( V ) S p ( W ) Sp ⁡ ( W ) × O ( V ) → S p ( W ) Sp(W)xxO(V)rarrSp(W)\operatorname{Sp}(W) \times \mathrm{O}(V) \rightarrow \mathrm{Sp}(\mathbb{W})Sp⁡(W)×O(V)→Sp(W)
then S p ( W ) S p ( W ) Sp(W)\mathrm{Sp}(W)Sp(W) and O ( V ) O ( V ) O(V)\mathrm{O}(V)O(V) are mutual commutants in S p ( W ) S p ( W ) Sp(W)\mathrm{Sp}(\mathbb{W})Sp(W).
Roughly speaking, the theta lifting is an integral transform with kernel given by a particular automorphic form on Sp ( W ) ( A ) Sp ⁡ ( W ) ( A ) Sp(W)(A)\operatorname{Sp}(\mathbb{W})(\mathbb{A})Sp⁡(W)(A) restricted to Sp ( W ) ( A ) × O ( V ) ( A ) Sp ⁡ ( W ) ( A ) × O ( V ) ( A ) Sp(W)(A)xxO(V)(A)\operatorname{Sp}(W)(\mathbb{A}) \times \mathrm{O}(V)(\mathbb{A})Sp⁡(W)(A)×O(V)(A). To be precise, we need to consider the metaplectic group M p ( W ) ( A ) M p ( W ) ( A ) Mp(W)(A)\mathrm{Mp}(\mathbb{W})(\mathbb{A})Mp(W)(A), which is a nontrivial topological central extension
1 { ± 1 } Mp ( W ) ( A ) Sp ( W ) ( A ) 1 1 → { ± 1 } → Mp ⁡ ( W ) ( A ) → Sp ⁡ ( W ) ( A ) → 1 1rarr{+-1}rarr Mp(W)(A)rarr Sp(W)(A)rarr11 \rightarrow\{ \pm 1\} \rightarrow \operatorname{Mp}(\mathbb{W})(\mathbb{A}) \rightarrow \operatorname{Sp}(\mathbb{W})(\mathbb{A}) \rightarrow 11→{±1}→Mp⁡(W)(A)→Sp⁡(W)(A)→1
(Here we have abused notation since Mp ( W ) ( A ) Mp ⁡ ( W ) ( A ) Mp(W)(A)\operatorname{Mp}(\mathbb{W})(\mathbb{A})Mp⁡(W)(A) is not the group of A A A\mathbb{A}A-valued points of an algebraic group over F F FFF.) This extension splits over Sp ( W ) ( F ) Sp ⁡ ( W ) ( F ) Sp(W)(F)\operatorname{Sp}(\mathbb{W})(F)Sp⁡(W)(F) canonically, so that we may speak of automorphic forms on Mp ( W ) ( A ) Mp ⁡ ( W ) ( A ) Mp(W)(A)\operatorname{Mp}(\mathbb{W})(\mathbb{A})Mp⁡(W)(A). We are interested in a particular representation ω ω omega\omegaω of Mp ( W ) ( A ) Mp ⁡ ( W ) ( A ) Mp(W)(A)\operatorname{Mp}(\mathbb{W})(\mathbb{A})Mp⁡(W)(A) (depending on a choice of a nontrivial additive character of A / F A / F A//F\mathbb{A} / FA/F ), called the Weil representation [61], which is a representation theoretic incarnation of theta functions. This representation has an automorphic realization, i.e., there is an Mp ( W ) ( A ) Mp ⁡ ( W ) ( A ) Mp(W)(A)\operatorname{Mp}(\mathbb{W})(\mathbb{A})Mp⁡(W)(A)-equivariant map φ θ φ map ⁡ φ ↦ θ φ map varphi|->theta_(varphi)\operatorname{map} \varphi \mapsto \theta_{\varphi}map⁡φ↦θφ from ω ω omega\omegaω to the space of automorphic forms on Mp ( W ) ( A ) Mp ⁡ ( W ) ( A ) Mp(W)(A)\operatorname{Mp}(\mathbb{W})(\mathbb{A})Mp⁡(W)(A). On the other hand, there exists a dotted arrow making the following diagram commute:
(Note that it descends to a homomorphism from the bottom left corner if and only if dim V dim ⁡ V dim V\operatorname{dim} Vdim⁡V is even.) Thus we may regard θ φ θ φ theta_(varphi)\theta_{\varphi}θφ as an automorphic form on Mp ( W ) ( A ) × O ( V ) ( A ) Mp ⁡ ( W ) ( A ) × O ( V ) ( A ) Mp(W)(A)xxO(V)(A)\operatorname{Mp}(W)(\mathbb{A}) \times \mathrm{O}(V)(\mathbb{A})Mp⁡(W)(A)×O(V)(A) by restriction and associate to an automorphic form f f fff on Mp ( W ) ( A ) Mp ⁡ ( W ) ( A ) Mp(W)(A)\operatorname{Mp}(W)(\mathbb{A})Mp⁡(W)(A) an automorphic form θ φ ( f ) θ φ ( f ) theta_(varphi)(f)\theta_{\varphi}(f)θφ(f) on O ( V ) ( A ) O ( V ) ( A ) O(V)(A)\mathrm{O}(V)(\mathbb{A})O(V)(A) by setting
θ φ ( f ) ( h ) = Sp ( W ) ( F ) Mp ( W ) ( A ) θ φ ( g , h ) f ( g ) ¯ d g θ φ ( f ) ( h ) = ∫ Sp ⁡ ( W ) ( F ) ∖ Mp ⁡ ( W ) ( A )   θ φ ( g , h ) f ( g ) ¯ d g theta_(varphi)(f)(h)=int_(Sp(W)(F)\\Mp(W)(A))theta_(varphi)(g,h) bar(f(g))dg\theta_{\varphi}(f)(h)=\int_{\operatorname{Sp}(W)(F) \backslash \operatorname{Mp}(W)(\mathbb{A})} \theta_{\varphi}(g, h) \overline{f(g)} d gθφ(f)(h)=∫Sp⁡(W)(F)∖Mp⁡(W)(A)θφ(g,h)f(g)¯dg
provided the integral converges, e.g., if f f fff is cuspidal.
For any irreducible cuspidal automorphic representation π Ï€ pi\piÏ€ of Mp ( W ) ( A ) Mp ⁡ ( W ) ( A ) Mp(W)(A)\operatorname{Mp}(W)(\mathbb{A})Mp⁡(W)(A), we define the theta lift θ ( π ) θ ( Ï€ ) theta(pi)\theta(\pi)θ(Ï€) of π Ï€ pi\piÏ€ as the automorphic representation of O ( V ) ( A ) O ( V ) ( A ) O(V)(A)O(V)(\mathbb{A})O(V)(A) spanned by θ φ ( f ) θ φ ( f ) theta_(varphi)(f)\theta_{\varphi}(f)θφ(f) for all φ ω φ ∈ ω varphi in omega\varphi \in \omegaφ∈ω and f π f ∈ Ï€ f in pif \in \pif∈π. We only consider the case when π Ï€ pi\piÏ€ descends (resp. does not descend)
to a representation of Sp ( W ) ( A ) Sp ⁡ ( W ) ( A ) Sp(W)(A)\operatorname{Sp}(W)(\mathbb{A})Sp⁡(W)(A) if dim V dim ⁡ V dim V\operatorname{dim} Vdim⁡V is even (resp. odd); otherwise θ ( π ) θ ( Ï€ ) theta(pi)\theta(\pi)θ(Ï€) is always zero. To describe θ ( π ) θ ( Ï€ ) theta(pi)\theta(\pi)θ(Ï€), we need to introduce the local analog of the theta lifting. First, note that the map ( φ , f ) θ φ ( f ) ( φ , f ) ↦ θ φ ( f ) (varphi,f)|->theta_(varphi)(f)(\varphi, f) \mapsto \theta_{\varphi}(f)(φ,f)↦θφ(f) defines an element in
Hom Mp ( W ) ( A ) × O ( V ) ( A ) ( ω π ¯ , θ ( π ) ) Hom Mp ( W ) ( A ) × O ( V ) ( A ) ( ω , π θ ( π ) ) Hom Mp ⁡ ( W ) ( A ) × O ( V ) ( A ) ⁡ ( ω ⊗ Ï€ ¯ , θ ( Ï€ ) ) ≅ Hom Mp ⁡ ( W ) ( A ) × O ( V ) ( A ) ⁡ ( ω , Ï€ ⊗ θ ( Ï€ ) ) Hom_(Mp(W)(A)xx O(V)(A))(omega ox bar(pi),theta(pi))~=Hom_(Mp(W)(A)xx O(V)(A))(omega,pi ox theta(pi))\operatorname{Hom}_{\operatorname{Mp}(W)(\mathbb{A}) \times O(V)(\mathbb{A})}(\omega \otimes \bar{\pi}, \theta(\pi)) \cong \operatorname{Hom}_{\operatorname{Mp}(W)(\mathbb{A}) \times O(V)(\mathbb{A})}(\omega, \pi \otimes \theta(\pi))HomMp⁡(W)(A)×O(V)(A)⁡(ω⊗π¯,θ(Ï€))≅HomMp⁡(W)(A)×O(V)(A)⁡(ω,π⊗θ(Ï€))
since π Ï€ pi\piÏ€ is unitary. Recall that ω ω omega\omegaω can be regarded as the restricted tensor product of the local Weil representations ω v ω v omega_(v)\omega_{v}ωv of Mp ( W ) ( F v ) Mp ⁡ ( W ) F v Mp(W)(F_(v))\operatorname{Mp}(\mathbb{W})\left(F_{v}\right)Mp⁡(W)(Fv) via the surjection v Mp ( W ) ( F v ) Mp ( W ) ( A ) ∏ v ′   Mp ⁡ ( W ) F v → Mp ⁡ ( W ) ( A ) prod_(v)^(')Mp(W)(F_(v))rarr Mp(W)(A)\prod_{v}^{\prime} \operatorname{Mp}(\mathbb{W})\left(F_{v}\right) \rightarrow \operatorname{Mp}(\mathbb{W})(\mathbb{A})∏v′Mp⁡(W)(Fv)→Mp⁡(W)(A), where Mp ( W ) ( F v ) Mp ⁡ ( W ) F v Mp(W)(F_(v))\operatorname{Mp}(\mathbb{W})\left(F_{v}\right)Mp⁡(W)(Fv) is the metaplectic cover of Sp ( W ) ( F v ) Sp ⁡ ( W ) F v Sp(W)(F_(v))\operatorname{Sp}(\mathbb{W})\left(F_{v}\right)Sp⁡(W)(Fv). Similarly, π Ï€ pi\piÏ€ can be decomposed as π v π v Ï€ ≅ ⊗ v Ï€ v pi~=ox_(v)pi_(v)\pi \cong \otimes_{v} \pi_{v}π≅⊗vÏ€v, where π v Ï€ v pi_(v)\pi_{v}Ï€v is an irreducible representation of Mp ( W ) ( F v ) Mp ⁡ ( W ) F v Mp(W)(F_(v))\operatorname{Mp}(W)\left(F_{v}\right)Mp⁡(W)(Fv). We define the local theta lift θ ( π v ) θ Ï€ v theta(pi_(v))\theta\left(\pi_{v}\right)θ(Ï€v) of π v Ï€ v pi_(v)\pi_{v}Ï€v as an irreducible representation of O ( V ) ( F v ) O ( V ) F v O(V)(F_(v))\mathrm{O}(V)\left(F_{v}\right)O(V)(Fv) such that
Hom Mp ( W ) ( F v ) × O ( V ) ( F v ) ( ω v , π v θ ( π v ) ) 0 Hom Mp ⁡ ( W ) F v × O ( V ) F v ⁡ ω v , Ï€ v ⊗ θ Ï€ v ≠ 0 Hom_(Mp(W)(F_(v))xx O(V)(F_(v)))(omega_(v),pi_(v)ox theta(pi_(v)))!=0\operatorname{Hom}_{\operatorname{Mp}(W)\left(F_{v}\right) \times O(V)\left(F_{v}\right)}\left(\omega_{v}, \pi_{v} \otimes \theta\left(\pi_{v}\right)\right) \neq 0HomMp⁡(W)(Fv)×O(V)(Fv)⁡(ωv,Ï€v⊗θ(Ï€v))≠0
which is unique (if it exists) by the Howe duality [ 23 , 31 , 55 ] [ 23 , 31 , 55 ] [23,31,55][23,31,55][23,31,55]. (When such a representation does not exist, we interpret θ ( π v ) θ Ï€ v theta(pi_(v))\theta\left(\pi_{v}\right)θ(Ï€v) as zero.) Now assume that θ ( π ) θ ( Ï€ ) theta(pi)\theta(\pi)θ(Ï€) is nonzero and cuspidal. Then it follows from the Howe duality that θ ( π ) θ ( Ï€ ) theta(pi)\theta(\pi)θ(Ï€) is irreducible and can be decomposed as
θ ( π ) v θ ( π v ) θ ( Ï€ ) ≅ ⊗ v θ Ï€ v theta(pi)~=ox_(v)theta(pi_(v))\theta(\pi) \cong \otimes_{v} \theta\left(\pi_{v}\right)θ(Ï€)≅⊗vθ(Ï€v)
Remark 2.1. We may extend the Weil representation and define the theta lifting for the pair ( GSp ( W ) , GO ( V ) ) ( GSp ⁡ ( W ) , GO ⁡ ( V ) ) (GSp(W),GO(V))(\operatorname{GSp}(W), \operatorname{GO}(V))(GSp⁡(W),GO⁡(V)), where GSp ( W ) GSp ⁡ ( W ) GSp(W)\operatorname{GSp}(W)GSp⁡(W) and GO ( V ) GO ⁡ ( V ) GO(V)\operatorname{GO}(V)GO⁡(V) are the similitude groups of W W WWW and V V VVV, respectively.

2.2. Explicit realization of the Jacquet-Langlands correspondence

From now on, we mainly consider the case when
dim W = 2 , dim V = 4 dim ⁡ W = 2 , dim ⁡ V = 4 dim W=2,quad dim V=4\operatorname{dim} W=2, \quad \operatorname{dim} V=4dim⁡W=2,dim⁡V=4
and the discriminant of V V VVV is trivial. Then we may identify W W WWW with the space F 2 F 2 F^(2)F^{2}F2, equipped with the form ( ( x 1 , x 2 ) , ( y 1 , y 2 ) ) W = x 1 y 2 x 2 y 1 x 1 , x 2 , y 1 , y 2 W = x 1 y 2 − x 2 y 1 ((x_(1),x_(2)),(y_(1),y_(2)))_(W)=x_(1)y_(2)-x_(2)y_(1)\left(\left(x_{1}, x_{2}\right),\left(y_{1}, y_{2}\right)\right)_{W}=x_{1} y_{2}-x_{2} y_{1}((x1,x2),(y1,y2))W=x1y2−x2y1, so that
GSp ( W ) = G L 2 GSp ⁡ ( W ) = G L 2 GSp(W)=GL_(2)\operatorname{GSp}(W)=\mathrm{GL}_{2}GSp⁡(W)=GL2
We may also identify V V VVV with a quaternion algebra B B BBB over F F FFF equipped with the form ( x , y ) V = Tr B / F ( x y ) ( x , y ) V = Tr B / F ⁡ x y ∗ (x,y)_(V)=Tr_(B//F)(xy^(**))(x, y)_{V}=\operatorname{Tr}_{B / F}\left(x y^{*}\right)(x,y)V=TrB/F⁡(xy∗), where Tr B / F Tr B / F Tr_(B//F)\operatorname{Tr}_{B / F}TrB/F is the reduced trace and ∗ ***∗ is the main involution, so that
GO ( V ) 0 = ( B × × B × ) / F × GO ⁡ ( V ) 0 = B × × B × / F × GO(V)^(0)=(B^(xx)xxB^(xx))//F^(xx)\operatorname{GO}(V)^{0}=\left(B^{\times} \times B^{\times}\right) / F^{\times}GO⁡(V)0=(B××B×)/F×
Here GO ( V ) 0 GO ⁡ ( V ) 0 GO(V)^(0)\operatorname{GO}(V)^{0}GO⁡(V)0 is the identity component of GO ( V ) GO ⁡ ( V ) GO(V)\operatorname{GO}(V)GO⁡(V) and B × × B × B × × B × B^(xx)xxB^(xx)B^{\times} \times B^{\times}B××B×acts on V V VVV by left and right multiplication.
Let π Ï€ pi\piÏ€ be an irreducible cuspidal automorphic representation of G L 2 ( A ) G L 2 ( A ) GL_(2)(A)\mathrm{GL}_{2}(\mathbb{A})GL2(A). We regard the theta lift θ ( π ) θ ( Ï€ ) theta(pi)\theta(\pi)θ(Ï€) of π Ï€ pi\piÏ€ (restricted to GO ( V ) 0 ( A ) GO ⁡ ( V ) 0 ( A ) GO(V)^(0)(A)\operatorname{GO}(V)^{0}(\mathbb{A})GO⁡(V)0(A) ) as an automorphic representation of B × ( A ) × B × ( A ) B × ( A ) × B × ( A ) B^(xx)(A)xxB^(xx)(A)B^{\times}(\mathbb{A}) \times B^{\times}(\mathbb{A})B×(A)×B×(A). Then Shimizu [49] proved that
θ ( π ) = π B π B θ ( Ï€ ) = Ï€ B ⊗ Ï€ B theta(pi)=pi^(B)oxpi^(B)\theta(\pi)=\pi^{B} \otimes \pi^{B}θ(Ï€)=Ï€B⊗πB
where π B Ï€ B pi^(B)\pi^{B}Ï€B is the Jacquet-Langlands transfer of π Ï€ pi\piÏ€ to B × ( A ) B × ( A ) B^(xx)(A)B^{\times}(\mathbb{A})B×(A). (When π Ï€ pi\piÏ€ does not transfer to B × ( A ) B × ( A ) B^(xx)(A)B^{\times}(\mathbb{A})B×(A), we interpret π B Ï€ B pi^(B)\pi^{B}Ï€B as zero.)
Remark 2.2. In [36], we gave the following variant of the above realization. Let B , B 1 , B 2 B , B 1 , B 2 B,B_(1),B_(2)B, B_{1}, B_{2}B,B1,B2 be three quaternion division algebras over F F FFF such that B = B 1 B 2 B = B 1 â‹… B 2 B=B_(1)*B_(2)B=B_{1} \cdot B_{2}B=B1â‹…B2 in the Brauer group. We consider a 1-dimensional Hermitian space W W WWW over B B BBB and a 2-dimensional skew-Hermitian space V V VVV over B B BBB such that
G U ( W ) = B × , G U ( V ) 0 = ( B 1 × × B 2 × ) / F × G U ( W ) = B × , G U ( V ) 0 = B 1 × × B 2 × / F × GU(W)=B^(xx),quadGU(V)^(0)=(B_(1)^(xx)xxB_(2)^(xx))//F^(xx)\mathrm{GU}(W)=B^{\times}, \quad \mathrm{GU}(V)^{0}=\left(B_{1}^{\times} \times B_{2}^{\times}\right) / F^{\times}GU(W)=B×,GU(V)0=(B1××B2×)/F×
where G U ( W ) G U ( W ) GU(W)\mathrm{GU}(W)GU(W) and G U ( V ) G U ( V ) GU(V)\mathrm{GU}(V)GU(V) are the unitary similitude groups of W W WWW and V V VVV, respectively. Let π B Ï€ B pi^(B)\pi^{B}Ï€B be an irreducible automorphic representation of B × ( A ) B × ( A ) B^(xx)(A)B^{\times}(\mathbb{A})B×(A) such that its Jacquet-Langlands transfer to G L 2 ( A ) G L 2 ( A ) GL_(2)(A)\mathrm{GL}_{2}(\mathbb{A})GL2(A) is cuspidal. Then we have
θ ( π B ) = π B 1 π B 2 θ Ï€ B = Ï€ B 1 ⊗ Ï€ B 2 theta(pi^(B))=pi^(B_(1))oxpi^(B_(2))\theta\left(\pi^{B}\right)=\pi^{B_{1}} \otimes \pi^{B_{2}}θ(Ï€B)=Ï€B1⊗πB2
where π B 1 Ï€ B 1 pi^(B_(1))\pi^{B_{1}}Ï€B1 and π B 2 Ï€ B 2 pi^(B_(2))\pi^{B_{2}}Ï€B2 are the Jacquet-Langlands transfers of π B Ï€ B pi^(B)\pi^{B}Ï€B to B 1 × ( A ) B 1 × ( A ) B_(1)^(xx)(A)B_{1}^{\times}(\mathbb{A})B1×(A) and B 2 × ( A ) B 2 × ( A ) B_(2)^(xx)(A)B_{2}^{\times}(\mathbb{A})B2×(A), respectively. We believe that this realization is useful to study integral period relations.

2.3. Seesaw identities

One of the advantages of the theta lifting is that it produces various period relations in a simple way, which was observed by Kudla [39]. Suppose that we have two reductive dual pairs ( G , H ) ( G , H ) (G,H)(G, H)(G,H) and ( G , H ) G ′ , H ′ (G^('),H^('))\left(G^{\prime}, H^{\prime}\right)(G′,H′) in the same symplectic group such that G G G ⊂ G ′ G subG^(')G \subset G^{\prime}G⊂G′ and H H H ⊃ H ′ H supH^(')H \supset H^{\prime}H⊃H′. This can be illustrated by the following picture, called a seesaw diagram:
Let f f fff and f f ′ f^(')f^{\prime}f′ be automorphic forms on G ( A ) G ( A ) G(A)G(\mathbb{A})G(A) and H ( A ) H ′ ( A ) H^(')(A)H^{\prime}(\mathbb{A})H′(A), respectively. Then the theta lifting produces automorphic forms θ φ ( f ) θ φ ( f ) theta_(varphi)(f)\theta_{\varphi}(f)θφ(f) and θ φ ( f ) θ φ f ′ theta_(varphi)(f^('))\theta_{\varphi}\left(f^{\prime}\right)θφ(f′) on H ( A ) H ( A ) H(A)H(\mathbb{A})H(A) and G ( A ) G ′ ( A ) G^(')(A)G^{\prime}(\mathbb{A})G′(A), respectively, and the so-called seesaw identity
θ φ ( f ) | H ( A ) , f = H ( F ) H ( A ) θ φ ( f ) ( h ) f ( h ) ¯ d h = G ( F ) G ( A ) H ( F ) H ( A ) θ φ ( g , h ) f ( g ) f ( h ) ¯ d g d h = G ( F ) G ( A ) f ( g ) ¯ θ φ ( f ) ( g ) d g = f , θ φ ( f ) | G ( A ) ) ¯ θ φ ( f ) H ′ ( A ) , f ′ = ∫ H ′ ( F ) ∖ H ′ ( A )   θ φ ( f ) h ′ f ′ h ′ ¯ d h ′ = ∫ G ( F ) ∖ G ( A )   ∫ H ′ ( F ) ∖ H ′ ( A )   θ φ g , h ′ f ( g ) f ′ h ′ ¯ d g d h ′ = ∫ G ( F ) ∖ G ( A )   f ( g ) ¯ θ φ f ′ ( g ) d g = f , θ φ f ′ G ( A ) ¯ {:[(:theta_(varphi)(f)|_(H^(')(A)),f^('):)=int_(H^(')(F)\\H^(')(A))theta_(varphi)(f)(h^(')) bar(f^(')(h^(')))dh^(')],[=int_(G(F)\\G(A))int_(H^(')(F)\\H^(')(A))theta_(varphi)(g,h^(')) bar(f(g)f^(')(h^(')))dgdh^(')],[=int_(G(F)\\G(A)) bar(f(g))theta_(varphi)(f^('))(g)dg= bar((:f,theta_(varphi)(f^('))|_(G(A))))]:}\begin{aligned} \left\langle\left.\theta_{\varphi}(f)\right|_{H^{\prime}(\mathbb{A})}, f^{\prime}\right\rangle & =\int_{H^{\prime}(F) \backslash H^{\prime}(\mathbb{A})} \theta_{\varphi}(f)\left(h^{\prime}\right) \overline{f^{\prime}\left(h^{\prime}\right)} d h^{\prime} \\ & =\int_{G(F) \backslash G(\mathbb{A})} \int_{H^{\prime}(F) \backslash H^{\prime}(\mathbb{A})} \theta_{\varphi}\left(g, h^{\prime}\right) \overline{f(g) f^{\prime}\left(h^{\prime}\right)} d g d h^{\prime} \\ & =\int_{G(F) \backslash G(\mathbb{A})} \overline{f(g)} \theta_{\varphi}\left(f^{\prime}\right)(g) d g=\overline{\left\langle f,\left.\theta_{\varphi}\left(f^{\prime}\right)\right|_{G(\mathbb{A})}\right)} \end{aligned}⟨θφ(f)|H′(A),f′⟩=∫H′(F)∖H′(A)θφ(f)(h′)f′(h′)¯dh′=∫G(F)∖G(A)∫H′(F)∖H′(A)θφ(g,h′)f(g)f′(h′)¯dgdh′=∫G(F)∖G(A)f(g)¯θφ(f′)(g)dg=⟨f,θφ(f′)|G(A))¯
provided the double integral converges absolutely. Here , ⟨ ⋅ , ⋅ ⟩ (:*,*:)\langle\cdot, \cdot\rangle⟨⋅,⋅⟩ denotes the Petersson inner product.
As an example of this identity, we recall Waldspurger's formula for torus periods. We keep the setup of the previous subsection. Fix a quadratic extension E E EEE of F F FFF which embeds into B B BBB and write B = E E j B = E ⊕ E j B=E o+EjB=E \oplus E jB=E⊕Ej with a trace zero element j j jjj in B B BBB. Let V = V 1 V 2 V = V 1 ⊕ V 2 V=V_(1)o+V_(2)V=V_{1} \oplus V_{2}V=V1⊕V2 be the corresponding decomposition of quadratic spaces, so that
GO ( V 1 ) 0 = G O ( V 2 ) 0 = E × GO ⁡ V 1 0 = G O V 2 0 = E × GO (V_(1))^(0)=GO(V_(2))^(0)=E^(xx)\operatorname{GO}\left(V_{1}\right)^{0}=\mathrm{GO}\left(V_{2}\right)^{0}=E^{\times}GO⁡(V1)0=GO(V2)0=E×
Then the identification W F V = ( W F V 1 ) ( W F V 2 ) W ⊗ F V = W ⊗ F V 1 ⊕ W ⊗ F V 2 Wox_(F)V=(Wox_(F)V_(1))o+(Wox_(F)V_(2))W \otimes_{F} V=\left(W \otimes_{F} V_{1}\right) \oplus\left(W \otimes_{F} V_{2}\right)W⊗FV=(W⊗FV1)⊕(W⊗FV2) gives rise to the following seesaw diagram:
Let π B Ï€ B pi^(B)\pi^{B}Ï€B be an irreducible automorphic representation of B × ( A ) B × ( A ) B^(xx)(A)B^{\times}(\mathbb{A})B×(A) such that its JacquetLanglands transfer π Ï€ pi\piÏ€ to G L 2 ( A ) G L 2 ( A ) GL_(2)(A)\mathrm{GL}_{2}(\mathbb{A})GL2(A) is cuspidal. Let χ χ chi\chiχ be an automorphic character of A E × A E × A_(E)^(xx)\mathbb{A}_{E}^{\times}AE×. Assume that the product of the central character of π B Ï€ B pi^(B)\pi^{B}Ï€B and the restriction of χ χ chi\chiχ to A F × A F × A_(F)^(xx)\mathbb{A}_{F}^{\times}AF×is trivial and consider the torus period
P ( f , χ ) = E × A F × A E × f ( h ) χ ( h ) d h P ( f , χ ) = ∫ E × A F × ∖ A E ×   f ( h ) χ ( h ) d h P(f,chi)=int_(E^(xx)A_(F)^(xx)\\A_(E)^(xx))f(h)chi(h)dhP(f, \chi)=\int_{E^{\times} \mathbb{A}_{F}^{\times} \backslash \mathbb{A}_{E}^{\times}} f(h) \chi(h) d hP(f,χ)=∫E×AF×∖AE×f(h)χ(h)dh
for a decomposable vector f π B f ∈ Ï€ B f inpi^(B)f \in \pi^{B}f∈πB. Then using the above seesaw diagram, Waldspurger [54] proved that
| P ( f , χ ) | 2 = 1 4 ζ ( 2 ) L ( 1 / 2 , π E × χ ) L ( 1 , π , Ad ) L ( 1 , μ E / F ) v α v ( f v , χ v ) | P ( f , χ ) | 2 = 1 4 ζ ( 2 ) L 1 / 2 , Ï€ E × χ L ( 1 , Ï€ , Ad ) L 1 , μ E / F ∏ v   α v f v , χ v |P(f,chi)|^(2)=(1)/(4)(zeta(2)L(1//2,pi_(E)xx chi))/(L(1,pi,Ad)L(1,mu_(E//F)))prod_(v)alpha_(v)(f_(v),chi_(v))|P(f, \chi)|^{2}=\frac{1}{4} \frac{\zeta(2) L\left(1 / 2, \pi_{E} \times \chi\right)}{L(1, \pi, \operatorname{Ad}) L\left(1, \mu_{E / F}\right)} \prod_{v} \alpha_{v}\left(f_{v}, \chi_{v}\right)|P(f,χ)|2=14ζ(2)L(1/2,Ï€E×χ)L(1,Ï€,Ad)L(1,μE/F)∏vαv(fv,χv)
where
  • ζ ( s ) ζ ( s ) zeta(s)\zeta(s)ζ(s) is the completed Dedekind zeta function of F F FFF,
  • L ( s , π E × χ ) L s , Ï€ E × χ L(s,pi_(E)xx chi)L\left(s, \pi_{E} \times \chi\right)L(s,Ï€E×χ) is the standard L L LLL-function of the base change π E Ï€ E pi_(E)\pi_{E}Ï€E of π Ï€ pi\piÏ€ to G L 2 ( A E ) G L 2 A E GL_(2)(A_(E))\mathrm{GL}_{2}\left(\mathbb{A}_{E}\right)GL2(AE) twisted by χ χ chi\chiχ,
  • L ( s , π L ( s , Ï€ L(s,piL(s, \piL(s,Ï€, Ad ) ) ))) is the adjoint L L LLL-function of π Ï€ pi\piÏ€,
  • L ( s , μ E / F ) L s , μ E / F L(s,mu_(E//F))L\left(s, \mu_{E / F}\right)L(s,μE/F) is the Hecke L L LLL-function of the quadratic automorphic character μ E / F μ E / F mu_(E//F)\mu_{E / F}μE/F of A F × A F × A_(F)^(xx)\mathbb{A}_{F}^{\times}AF×associated to E / F E / F E//FE / FE/F by class field theory,
  • α v ( f v , χ v ) α v f v , χ v alpha_(v)(f_(v),chi_(v))\alpha_{v}\left(f_{v}, \chi_{v}\right)αv(fv,χv) is a certain normalized local integral of matrix coefficients.
As another example, we consider the 6-dimensional symplectic space W = W 3 W ′ = W 3 W^(')=W^(3)W^{\prime}=W^{3}W′=W3 over F F FFF. Then the identification W F V = ( W F V ) 3 W ′ ⊗ F V = W ⊗ F V 3 W^(')ox_(F)V=(Wox_(F)V)^(3)W^{\prime} \otimes_{F} V=\left(W \otimes_{F} V\right)^{3}W′⊗FV=(W⊗FV)3 gives rise to the following seesaw diagram:
Let π 1 B , π 2 B , π 3 B Ï€ 1 B , Ï€ 2 B , Ï€ 3 B pi_(1)^(B),pi_(2)^(B),pi_(3)^(B)\pi_{1}^{B}, \pi_{2}^{B}, \pi_{3}^{B}Ï€1B,Ï€2B,Ï€3B be irreducible automorphic representations of B × ( A ) B × ( A ) B^(xx)(A)B^{\times}(\mathbb{A})B×(A) such that their Jacquet-Langlands transfers π 1 , π 2 , π 3 Ï€ 1 , Ï€ 2 , Ï€ 3 pi_(1),pi_(2),pi_(3)\pi_{1}, \pi_{2}, \pi_{3}Ï€1,Ï€2,Ï€3 to G L 2 ( A ) G L 2 ( A ) GL_(2)(A)\mathrm{GL}_{2}(\mathbb{A})GL2(A) are cuspidal. Assume that the product of the central characters of π 1 B , π 2 B , π 3 B Ï€ 1 B , Ï€ 2 B , Ï€ 3 B pi_(1)^(B),pi_(2)^(B),pi_(3)^(B)\pi_{1}^{B}, \pi_{2}^{B}, \pi_{3}^{B}Ï€1B,Ï€2B,Ï€3B is trivial and consider the trilinear period
P ( f 1 , f 2 , f 3 ) = B × A × B × ( A ) f 1 ( h ) f 2 ( h ) f 3 ( h ) d h P f 1 , f 2 , f 3 = ∫ B × A × ∖ B × ( A )   f 1 ( h ) f 2 ( h ) f 3 ( h ) d h P(f_(1),f_(2),f_(3))=int_(B^(xx)A^(xx)\\B^(xx)(A))f_(1)(h)f_(2)(h)f_(3)(h)dhP\left(f_{1}, f_{2}, f_{3}\right)=\int_{B^{\times} \mathbb{A}^{\times} \backslash B^{\times}(\mathbb{A})} f_{1}(h) f_{2}(h) f_{3}(h) d hP(f1,f2,f3)=∫B×A×∖B×(A)f1(h)f2(h)f3(h)dh
for decomposable vectors f 1 π 1 B , f 2 π 2 B , f 3 π 3 B f 1 ∈ Ï€ 1 B , f 2 ∈ Ï€ 2 B , f 3 ∈ Ï€ 3 B f_(1)inpi_(1)^(B),f_(2)inpi_(2)^(B),f_(3)inpi_(3)^(B)f_{1} \in \pi_{1}^{B}, f_{2} \in \pi_{2}^{B}, f_{3} \in \pi_{3}^{B}f1∈π1B,f2∈π2B,f3∈π3B. Then following the work of HarrisKudla [27] and using the above seesaw diagram, we proved in [32] that
| P ( f 1 , f 2 , f 3 ) | 2 = 1 8 ζ ( 2 ) 2 L ( 1 / 2 , π 1 × π 2 × π 3 ) L ( 1 , π 1 , Ad ) L ( 1 , π 2 , Ad ) L ( 1 , π 3 , Ad ) v α v ( f 1 , v , f 2 , v , f 3 , v ) P f 1 , f 2 , f 3 2 = 1 8 ζ ( 2 ) 2 L 1 / 2 , Ï€ 1 × Ï€ 2 × Ï€ 3 L 1 , Ï€ 1 , Ad L 1 , Ï€ 2 , Ad L 1 , Ï€ 3 , Ad ∏ v   α v f 1 , v , f 2 , v , f 3 , v |P(f_(1),f_(2),f_(3))|^(2)=(1)/(8)(zeta(2)^(2)L(1//2,pi_(1)xxpi_(2)xxpi_(3)))/(L(1,pi_(1),Ad)L(1,pi_(2),Ad)L(1,pi_(3),Ad))prod_(v)alpha_(v)(f_(1,v),f_(2,v),f_(3,v))\left|P\left(f_{1}, f_{2}, f_{3}\right)\right|^{2}=\frac{1}{8} \frac{\zeta(2)^{2} L\left(1 / 2, \pi_{1} \times \pi_{2} \times \pi_{3}\right)}{L\left(1, \pi_{1}, \operatorname{Ad}\right) L\left(1, \pi_{2}, \operatorname{Ad}\right) L\left(1, \pi_{3}, \operatorname{Ad}\right)} \prod_{v} \alpha_{v}\left(f_{1, v}, f_{2, v}, f_{3, v}\right)|P(f1,f2,f3)|2=18ζ(2)2L(1/2,Ï€1×π2×π3)L(1,Ï€1,Ad)L(1,Ï€2,Ad)L(1,Ï€3,Ad)∏vαv(f1,v,f2,v,f3,v)
where
  • L ( s , π 1 × π 2 × π 3 ) L s , Ï€ 1 × Ï€ 2 × Ï€ 3 L(s,pi_(1)xxpi_(2)xxpi_(3))L\left(s, \pi_{1} \times \pi_{2} \times \pi_{3}\right)L(s,Ï€1×π2×π3) is the triple product L L LLL-function of π 1 , π 2 , π 3 Ï€ 1 , Ï€ 2 , Ï€ 3 pi_(1),pi_(2),pi_(3)\pi_{1}, \pi_{2}, \pi_{3}Ï€1,Ï€2,Ï€3,
  • α v ( f 1 , v , f 2 , v , f 3 , v ) α v f 1 , v , f 2 , v , f 3 , v alpha_(v)(f_(1,v),f_(2,v),f_(3,v))\alpha_{v}\left(f_{1, v}, f_{2, v}, f_{3, v}\right)αv(f1,v,f2,v,f3,v) is a certain normalized local integral of matrix coefficients.
Remark 2.3. The above two formulas are special cases of the Gross-Prasad conjecture [25,26] and its refinement [33]. This conjecture (for special orthogonal groups) was extended to all classical groups by Gan-Gross-Prasad [17], and after the breakthrough of Zhang [63,64], the global conjecture for unitary groups has been proved in a series of works [8-11,62] using the relative trace formula. We should also mention the stunning work of Waldspurger [57-60], which led to the proof of the local Gan-Gross-Prasad conjecture for Bessel models [5-7,45] and Fourier-Jacobi models [2,19] in the p p ppp-adic case, where the theta lifting is used to deduce the latter from the former.

3. THE SHIMURA-WALDSPURGER CORRESPONDENCE

In this section, we review some applications of the theta lifting to automorphic forms on metaplectic groups.

3.1. Modular forms of half-integral weight

The theta function
θ ( z ) = n = e 2 π i n 2 z θ ( z ) = ∑ n = − ∞ ∞   e 2 Ï€ i n 2 z theta(z)=sum_(n=-oo)^(oo)e^(2pi in^(2)z)\theta(z)=\sum_{n=-\infty}^{\infty} e^{2 \pi i n^{2} z}θ(z)=∑n=−∞∞e2Ï€in2z
is a modular form of weight 1 / 2 1 / 2 1//21 / 21/2 and its significance is well known. Thus it is natural to study modular forms of half-integral weight, but Hecke [28, P. 152] realized the difficulty in developing the arithmetic theory; the Hecke operator T n T n T_(n)T_{n}Tn is zero unless n n nnn is a square. In 1973, Shimura [50] revolutionized the theory of modular forms of half-integral weight by relating them to modular forms of integral weight, i.e., he constructed a modular form of weight 2 k 2 k 2k2 k2k from a cusp form of weight k + 1 / 2 k + 1 / 2 k+1//2k+1 / 2k+1/2 by using the converse theorem, where k k kkk is a positive integer. Soon after the discovery of this correspondence, Niwa [46] and Shintani [51] gave an alternative construction using the theta lifting. This was further investigated by Waldspurger [53,56] in the framework of automorphic representations. Namely, he established a correspondence between automorphic representations of M p 2 ( A ) M p 2 ( A ) Mp_(2)(A)\mathrm{Mp}_{2}(\mathbb{A})Mp2(A) (where M p 2 ( A ) M p 2 ( A ) Mp_(2)(A)\mathrm{Mp}_{2}(\mathbb{A})Mp2(A) is the metaplectic cover of S L 2 ( A ) ) S L 2 ( A ) {:SL_(2)(A))\left.\mathrm{SL}_{2}(\mathbb{A})\right)SL2(A)) and those of P G L 2 ( A ) P G L 2 ( A ) PGL_(2)(A)\mathrm{PGL}_{2}(\mathbb{A})PGL2(A), which can be viewed as an example of Langlands functoriality.

3.2. Global correspondence

Now we discuss a generalization of the Shimura-Waldspurger correspondence to metaplectic groups of higher rank. Let F F FFF be a number field with adèle ring A A A\mathbb{A}A. We denote by S p 2 n S p 2 n Sp_(2n)\mathrm{Sp}_{2 n}Sp2n the symplectic group of rank n n nnn over F F FFF and by Mp 2 n ( A ) Mp 2 n ⁡ ( A ) Mp_(2n)(A)\operatorname{Mp}_{2 n}(\mathbb{A})Mp2n⁡(A) the metaplectic cover of S p 2 n ( A ) S p 2 n ( A ) Sp_(2n)(A)\mathrm{Sp}_{2 n}(\mathbb{A})Sp2n(A). Recall that this cover splits over S p 2 n ( F ) S p 2 n ( F ) Sp_(2n)(F)\mathrm{Sp}_{2 n}(F)Sp2n(F) canonically, so that we may speak of the unitary representation of Mp 2 n ( A ) Mp 2 n ⁡ ( A ) Mp_(2n)(A)\operatorname{Mp}_{2 n}(\mathbb{A})Mp2n⁡(A) on the Hilbert space
L 2 ( Sp 2 n ( F ) Mp 2 n ( A ) ) L 2 Sp 2 n ⁡ ( F ) ∖ Mp 2 n ⁡ ( A ) L^(2)(Sp_(2n)(F)\\Mp_(2n)(A))L^{2}\left(\operatorname{Sp}_{2 n}(F) \backslash \operatorname{Mp}_{2 n}(\mathbb{A})\right)L2(Sp2n⁡(F)∖Mp2n⁡(A))
given by right translation. Since we are interested in genuine automorphic representations of Mp 2 n ( A ) Mp 2 n ⁡ ( A ) Mp_(2n)(A)\operatorname{Mp}_{2 n}(\mathbb{A})Mp2n⁡(A), i.e., those which do not descend to representations of S p 2 n ( A ) S p 2 n ( A ) Sp_(2n)(A)\mathrm{Sp}_{2 n}(\mathbb{A})Sp2n(A), we only consider its subspace
L 2 ( M p 2 n ) L 2 M p 2 n L^(2)(Mp_(2n))L^{2}\left(\mathrm{Mp}_{2 n}\right)L2(Mp2n)
on which the central subgroup { ± 1 } { ± 1 } {+-1}\{ \pm 1\}{±1} acts by the nontrivial character. Write
L 2 ( M p 2 n ) = L disc 2 ( M p 2 n ) L cont 2 ( M p 2 n ) L 2 M p 2 n = L disc  2 M p 2 n ⊕ L cont  2 M p 2 n L^(2)(Mp_(2n))=L_("disc ")^(2)(Mp_(2n))o+L_("cont ")^(2)(Mp_(2n))L^{2}\left(\mathrm{Mp}_{2 n}\right)=L_{\text {disc }}^{2}\left(\mathrm{Mp}_{2 n}\right) \oplus L_{\text {cont }}^{2}\left(\mathrm{Mp}_{2 n}\right)L2(Mp2n)=Ldisc 2(Mp2n)⊕Lcont 2(Mp2n)
for the decomposition into the discrete part and the continuous part. Then the theory of Eisenstein series gives an explicit description of L cont 2 ( M p 2 n ) L cont  2 M p 2 n L_("cont ")^(2)(Mp_(2n))L_{\text {cont }}^{2}\left(\mathrm{Mp}_{2 n}\right)Lcont 2(Mp2n) in terms of automorphic discrete spectra of proper Levi subgroups of M p 2 n M p 2 n Mp_(2n)\mathrm{Mp}_{2 n}Mp2n, i.e., G L n 1 × × G L n k × M p 2 n 0 G L n 1 × ⋯ × G L n k × M p 2 n 0 GL_(n_(1))xx cdots xxGL_(n_(k))xxMp_(2n_(0))\mathrm{GL}_{n_{1}} \times \cdots \times \mathrm{GL}_{n_{k}} \times \mathrm{Mp}_{2 n_{0}}GLn1×⋯×GLnk×Mp2n0 with n 1 + + n k + n 0 = n n 1 + ⋯ + n k + n 0 = n n_(1)+cdots+n_(k)+n_(0)=nn_{1}+\cdots+n_{k}+n_{0}=nn1+⋯+nk+n0=n and n 0 < n n 0 < n n_(0) < nn_{0}<nn0<n. Thus the problem is to describe the irreducible decomposition of L disc 2 ( M p 2 n ) L disc  2 M p 2 n L_("disc ")^(2)(Mp_(2n))L_{\text {disc }}^{2}\left(\mathrm{Mp}_{2 n}\right)Ldisc 2(Mp2n).
To attack this problem, it is better to divide it into two parts as follows:
(1) Describe the decomposition of L disc 2 ( M p 2 n ) L disc  2 M p 2 n L_("disc ")^(2)(Mp_(2n))L_{\text {disc }}^{2}\left(\mathrm{Mp}_{2 n}\right)Ldisc 2(Mp2n) into near equivalence classes. Here we say that two irreducible genuine representations π v π v Ï€ ≅ ⊗ v Ï€ v pi~=ox_(v)pi_(v)\pi \cong \otimes_{v} \pi_{v}π≅⊗vÏ€v and π v π v Ï€ ′ ≅ ⊗ v Ï€ v ′ pi^(')~=ox_(v)pi_(v)^(')\pi^{\prime} \cong \otimes_{v} \pi_{v}^{\prime}π′≅⊗vÏ€v′ of Mp 2 n ( A ) Mp 2 n ⁡ ( A ) Mp_(2n)(A)\operatorname{Mp}_{2 n}(\mathbb{A})Mp2n⁡(A) are nearly equivalent if π v Ï€ v pi_(v)\pi_{v}Ï€v and π v Ï€ v ′ pi_(v)^(')\pi_{v}^{\prime}Ï€v′ are equivalent for almost all places v v vvv of F F FFF. (In particular, if π Ï€ pi\piÏ€ and π Ï€ ′ pi^(')\pi^{\prime}π′ are equivalent, then they are nearly equivalent.) Note that π v Ï€ v pi_(v)\pi_{v}Ï€v is unramified for almost all v v vvv, so that it determines and is determined by a semisimple conjugacy class c ψ v ( π v ) c ψ v Ï€ v c_(psi_(v))(pi_(v))c_{\psi_{v}}\left(\pi_{v}\right)cψv(Ï€v) in S p 2 n ( C ) S p 2 n ( C ) Sp_(2n)(C)\mathrm{Sp}_{2 n}(\mathbb{C})Sp2n(C) (depending on a choice of a nontrivial additive character ψ v ψ v psi_(v)\psi_{v}ψv of F v F v F_(v)F_{v}Fv ) via the Satake isomorphism. In other words, the near equivalence classes of irreducible genuine representations of M p 2 n ( A ) M p 2 n ( A ) Mp_(2n)(A)\mathrm{Mp}_{2 n}(\mathbb{A})Mp2n(A) can be parametrized by families of semisimple conjugacy classes
{ c v } v c v v {c_(v)}_(v)\left\{c_{v}\right\}_{v}{cv}v
in S p 2 n ( C ) S p 2 n ( C ) Sp_(2n)(C)\mathrm{Sp}_{2 n}(\mathbb{C})Sp2n(C), where we identify two families if they are equal for almost all v v vvv. Thus we want to describe the families { c v } v c v v {c_(v)}_(v)\left\{c_{v}\right\}_{v}{cv}v which correspond to the near equivalence classes in L disc 2 ( M p 2 n ) L disc  2 M p 2 n L_("disc ")^(2)(Mp_(2n))L_{\text {disc }}^{2}\left(\mathrm{Mp}_{2 n}\right)Ldisc 2(Mp2n).
(2) Describe the irreducible decomposition of each near equivalence class. Namely, for any near equivalence class C C CCC in L disc 2 ( M p 2 n ) L disc  2 M p 2 n L_("disc ")^(2)(Mp_(2n))L_{\text {disc }}^{2}\left(\mathrm{Mp}_{2 n}\right)Ldisc 2(Mp2n) and any irreducible genuine representation π Ï€ pi\piÏ€ of Mp 2 n ( A ) Mp 2 n ⁡ ( A ) Mp_(2n)(A)\operatorname{Mp}_{2 n}(\mathbb{A})Mp2n⁡(A), we want to give an explicit formula for the multiplicity of π Ï€ pi\piÏ€ in C C CCC in terms of the classification of representations.
In [20], we solved (1) completely and (2) partially; we described the families { c v } v c v v {c_(v)}_(v)\left\{c_{v}\right\}_{v}{cv}v as above in terms of automorphic representations of general linear groups, and admitting that Arthur's endoscopic classification [1] can be extended to nonsplit odd special orthogonal groups, we established the multiplicity formula for the tempered part of L disc 2 ( M p 2 n ) L disc  2 M p 2 n L_("disc ")^(2)(Mp_(2n))L_{\text {disc }}^{2}\left(\mathrm{Mp}_{2 n}\right)Ldisc 2(Mp2n).
Now we state the first result precisely.
Theorem 3.1 ([20]). Fix a nontrivial additive character ψ = v ψ v ψ = ⊗ v ψ v psi=ox_(v)psi_(v)\psi=\otimes_{v} \psi_{v}ψ=⊗vψv of A / F A / F A//F\mathbb{A} / FA/F. Then we have a decomposition
L disc 2 ( M p 2 n ) = ϕ L ϕ 2 ( M p 2 n ) L disc  2 M p 2 n = ⨁ Ï•   L Ï• 2 M p 2 n L_("disc ")^(2)(Mp_(2n))=bigoplus_(phi)L_(phi)^(2)(Mp_(2n))L_{\text {disc }}^{2}\left(\mathrm{Mp}_{2 n}\right)=\bigoplus_{\phi} L_{\phi}^{2}\left(\mathrm{Mp}_{2 n}\right)Ldisc 2(Mp2n)=⨁ϕLÏ•2(Mp2n)
where ϕ Ï• phi\phiÏ• runs over elliptic A-parameters for M p 2 n M p 2 n Mp_(2n)\mathrm{Mp}_{2 n}Mp2n. Here an elliptic A-parameter for M p 2 n M p 2 n Mp_(2n)\mathrm{Mp}_{2 n}Mp2n is defined to be a formal finite direct sum
i ϕ i S d i ⨁ i   Ï• i ⊗ S d i bigoplus_(i)phi_(i)oxS_(d_(i))\bigoplus_{i} \phi_{i} \otimes S_{d_{i}}⨁iÏ•i⊗Sdi
(which is a substitute for a hypothetical L-homomorphism L F × S L 2 ( C ) S p 2 n ( C ) L F × S L 2 ( C ) → S p 2 n ( C ) L_(F)xxSL_(2)(C)rarrSp_(2n)(C)\mathscr{L}_{F} \times \mathrm{SL}_{2}(\mathbb{C}) \rightarrow \mathrm{Sp}_{2 n}(\mathbb{C})LF×SL2(C)→Sp2n(C) ), where
  • ϕ i Ï• i phi_(i)\phi_{i}Ï•i is an irreducible self-dual cuspidal automorphic representation of G L n i ( A ) G L n i ( A ) GL_(n_(i))(A)\mathrm{GL}_{n_{i}}(\mathbb{A})GLni(A) (which is hypothetically identified with an n i n i n_(i)n_{i}ni-dimensional irreducible representation of L F L F L_(F)\mathscr{L}_{F}LF ),
  • S d i S d i S_(d_(i))S_{d_{i}}Sdi is the d i d i d_(i)d_{i}di-dimensional irreducible representation of S L 2 ( C ) S L 2 ( C ) SL_(2)(C)\mathrm{SL}_{2}(\mathbb{C})SL2(C),
  • if d i d i d_(i)d_{i}di is odd, then ϕ i Ï• i phi_(i)\phi_{i}Ï•i is symplectic, i.e., the exterior square L L LLL-function L ( s , ϕ i , 2 ) L s , Ï• i , ∧ 2 L(s,phi_(i),^^^(2))L\left(s, \phi_{i}, \wedge^{2}\right)L(s,Ï•i,∧2) has a pole at s = 1 s = 1 s=1s=1s=1 (and hence n i n i n_(i)n_{i}ni is even),
  • if d i d i d_(i)d_{i}di is even, then ϕ i Ï• i phi_(i)\phi_{i}Ï•i is orthogonal, i.e., the symmetric square L-function L ( s , ϕ i L s , Ï• i L(s,phi_(i):}L\left(s, \phi_{i}\right.L(s,Ï•i, Sym 2 ) 2 {:^(2))\left.^{2}\right)2) has a pole at s = 1 s = 1 s=1s=1s=1,
  • if i j i ≠ j i!=ji \neq ji≠j, then ( ϕ i , d i ) ( ϕ j , d j ) Ï• i , d i ≠ Ï• j , d j (phi_(i),d_(i))!=(phi_(j),d_(j))\left(\phi_{i}, d_{i}\right) \neq\left(\phi_{j}, d_{j}\right)(Ï•i,di)≠(Ï•j,dj),
  • i n i d i = 2 n ∑ i   n i d i = 2 n sum_(i)n_(i)d_(i)=2n\sum_{i} n_{i} d_{i}=2 n∑inidi=2n.
Also, L ϕ 2 ( M p 2 n ) L Ï• 2 M p 2 n L_(phi)^(2)(Mp_(2n))L_{\phi}^{2}\left(\mathrm{Mp}_{2 n}\right)LÏ•2(Mp2n) is defined as the near equivalence class in L disc 2 ( M p 2 n ) L disc  2 M p 2 n L_("disc ")^(2)(Mp_(2n))L_{\text {disc }}^{2}\left(\mathrm{Mp}_{2 n}\right)Ldisc 2(Mp2n) which corresponds to the family of semisimple conjugacy classes
{ c v ( ϕ v ) } v c v Ï• v v {c_(v)(phi_(v))}_(v)\left\{c_{v}\left(\phi_{v}\right)\right\}_{v}{cv(Ï•v)}v
in S p 2 n ( C ) S p 2 n ( C ) Sp_(2n)(C)\mathrm{Sp}_{2 n}(\mathbb{C})Sp2n(C) given as follows (so that any irreducible summand π Ï€ pi\piÏ€ of L ϕ 2 ( M p 2 n ) L Ï• 2 M p 2 n L_(phi)^(2)(Mp_(2n))L_{\phi}^{2}\left(\mathrm{Mp}_{2 n}\right)LÏ•2(Mp2n) satisfies c ψ v ( π v ) = c v ( ϕ v ) c ψ v Ï€ v = c v Ï• v c_(psi_(v))(pi_(v))=c_(v)(phi_(v))c_{\psi_{v}}\left(\pi_{v}\right)=c_{v}\left(\phi_{v}\right)cψv(Ï€v)=cv(Ï•v) for almost all v v vvv ). Suppose that v v vvv is finite and ϕ i , v Ï• i , v phi_(i,v)\phi_{i, v}Ï•i,v is unramified for all i i iii. Let c v ( ϕ i , v ) c v Ï• i , v c_(v)(phi_(i,v))c_{v}\left(\phi_{i, v}\right)cv(Ï•i,v) be the semisimple conjugacy class in G L n i ( C ) G L n i ( C ) GL_(n_(i))(C)\mathrm{GL}_{n_{i}}(\mathbb{C})GLni(C) which corresponds to ϕ i , v Ï• i , v phi_(i,v)\phi_{i, v}Ï•i,v and put
Q v ( d ) = ( q v ( d 1 ) / 2 q v ( d 3 ) / 2 q v ( d 1 ) / 2 ) Q v ( d ) = q v ( d − 1 ) / 2 q v ( d − 3 ) / 2 ⋱ q v − ( d − 1 ) / 2 Q_(v)(d)=([q_(v)^((d-1)//2),,,],[,q_(v)^((d-3)//2),,],[,,ddots,],[,,,q_(v)^(-(d-1)//2)])Q_{v}(d)=\left(\begin{array}{llll} q_{v}^{(d-1) / 2} & & & \\ & q_{v}^{(d-3) / 2} & & \\ & & \ddots & \\ & & & q_{v}^{-(d-1) / 2} \end{array}\right)Qv(d)=(qv(d−1)/2qv(d−3)/2⋱qv−(d−1)/2)
for any positive integer d d ddd, where q v q v q_(v)q_{v}qv is the cardinality of the residue field of F v F v F_(v)F_{v}Fv. We regard c v ( ϕ i , v ) Q v ( d i ) c v Ï• i , v ⊗ Q v d i c_(v)(phi_(i,v))oxQ_(v)(d_(i))c_{v}\left(\phi_{i, v}\right) \otimes Q_{v}\left(d_{i}\right)cv(Ï•i,v)⊗Qv(di) as a semisimple conjugacy class in S p n i d i ( C ) S p n i d i ( C ) Sp_(n_(i)d_(i))(C)\mathrm{Sp}_{n_{i} d_{i}}(\mathbb{C})Spnidi(C). Then we set
c v ( ϕ v ) = i c v ( ϕ i , v ) Q v ( d i ) c v Ï• v = ⨁ i   c v Ï• i , v ⊗ Q v d i c_(v)(phi_(v))=bigoplus_(i)c_(v)(phi_(i,v))oxQ_(v)(d_(i))c_{v}\left(\phi_{v}\right)=\bigoplus_{i} c_{v}\left(\phi_{i, v}\right) \otimes Q_{v}\left(d_{i}\right)cv(Ï•v)=⨁icv(Ï•i,v)⊗Qv(di)
To state the second result precisely, we need to introduce more notation. For each place v v vvv of F F FFF, let W F v W F v W_(F_(v))\mathcal{W}_{F_{v}}WFv be the Weil group of F v F v F_(v)F_{v}Fv and put
L F v = { W F v if v is infinite W F v × S L 2 ( C ) if v is finite L F v = W F v  if  v  is infinite  W F v × S L 2 ( C )  if  v  is finite  L_(F_(v))={[W_(F_(v))," if "v" is infinite "],[W_(F_(v))xxSL_(2)(C)," if "v" is finite "]:}\mathscr{L}_{F_{v}}= \begin{cases}\mathcal{W}_{F_{v}} & \text { if } v \text { is infinite } \\ \mathcal{W}_{F_{v}} \times \mathrm{SL}_{2}(\mathbb{C}) & \text { if } v \text { is finite }\end{cases}LFv={WFv if v is infinite WFv×SL2(C) if v is finite 
Let ϕ = i ϕ i S d i Ï• = ⨁ i   Ï• i ⊗ S d i phi=bigoplus_(i)phi_(i)oxS_(d_(i))\phi=\bigoplus_{i} \phi_{i} \otimes S_{d_{i}}Ï•=⨁iÏ•i⊗Sdi be an elliptic A A AAA-parameter for M p 2 n M p 2 n Mp_(2n)\mathrm{Mp}_{2 n}Mp2n. We regard its local component ϕ v = i ϕ i , v S d i Ï• v = ⨁ i   Ï• i , v ⊗ S d i phi_(v)=bigoplus_(i)phi_(i,v)oxS_(d_(i))\phi_{v}=\bigoplus_{i} \phi_{i, v} \otimes S_{d_{i}}Ï•v=⨁iÏ•i,v⊗Sdi at v v vvv as a local A A AAA-parameter
ϕ v : L F v × S L 2 ( C ) S p 2 n ( C ) Ï• v : L F v × S L 2 ( C ) → S p 2 n ( C ) phi_(v):L_(F_(v))xxSL_(2)(C)rarrSp_(2n)(C)\phi_{v}: \mathscr{L}_{F_{v}} \times \mathrm{SL}_{2}(\mathbb{C}) \rightarrow \mathrm{Sp}_{2 n}(\mathbb{C})Ï•v:LFv×SL2(C)→Sp2n(C)
via the local Langlands correspondence. Note that
c v ( ϕ v ) = ϕ v ( Fr v , ( q v 1 / 2 q v 1 / 2 ) ) c v Ï• v = Ï• v Fr v , q v 1 / 2 q v − 1 / 2 c_(v)(phi_(v))=phi_(v)(Fr_(v),([q_(v)^(1//2),],[,q_(v)^(-1//2)]))c_{v}\left(\phi_{v}\right)=\phi_{v}\left(\operatorname{Fr}_{v},\left(\begin{array}{ll} q_{v}^{1 / 2} & \\ & q_{v}^{-1 / 2} \end{array}\right)\right)cv(Ï•v)=Ï•v(Frv,(qv1/2qv−1/2))
for almost all v v vvv, where Fr v Fr v Fr_(v)\operatorname{Fr}_{v}Frv is a Frobenius element at v v vvv. We denote by ϕ v â„‘ Ï• v â„‘_(phi_(v))\Im_{\phi_{v}}â„‘Ï•v the component group of the centralizer of ϕ v Ï• v phi_(v)\phi_{v}Ï•v in S p 2 n ( C ) S p 2 n ( C ) Sp_(2n)(C)\mathrm{Sp}_{2 n}(\mathbb{C})Sp2n(C), which is an elementary abelian 2-group, and by
ς ϕ = i ( Z / 2 Z ) a i Ï‚ Ï• = ⨁ i   ( Z / 2 Z ) a i Ï‚_(phi)=bigoplus_(i)(Z//2Z)a_(i)\varsigma_{\phi}=\bigoplus_{i}(\mathbb{Z} / 2 \mathbb{Z}) a_{i}ςϕ=⨁i(Z/2Z)ai
the global component group of ϕ Ï• phi\phiÏ•, which is formally defined as an elementary abelian 2-group with a basis { a i } i a i i {a_(i)}_(i)\left\{a_{i}\right\}_{i}{ai}i indexed by { ϕ i S d i } i Ï• i ⊗ S d i i {phi_(i)oxS_(d_(i))}_(i)\left\{\phi_{i} \otimes S_{d_{i}}\right\}_{i}{Ï•i⊗Sdi}i. Then we have a natural homomorphism S ϕ S ϕ v S Ï• → S Ï• v S_(phi)rarrS_(phi_(v))S_{\phi} \rightarrow S_{\phi_{v}}Sϕ→SÏ•v for all v v vvv. We also consider the compact abelian group S ϕ , A = v ς ϕ v S Ï• , A = ∏ v   Ï‚ Ï• v S_(phi,A)=prod_(v)Ï‚_(phi_(v))S_{\phi, \mathbb{A}}=\prod_{v} \varsigma_{\phi_{v}}SÏ•,A=∏vςϕv and the diagonal map
Δ : S ϕ S ϕ , A Δ : S Ï• → S Ï• , A Delta:S_(phi)rarrS_(phi,A)\Delta: S_{\phi} \rightarrow S_{\phi, \mathbb{A}}Δ:Sϕ→SÏ•,A
Theorem 3.2 ([20]). Assume that ϕ Ï• phi\phiÏ• is tempered, i.e., d i = 1 d i = 1 d_(i)=1d_{i}=1di=1 for all i i iii. Then we have a decomposition
L ϕ 2 ( M p 2 n ) η m η π η L Ï• 2 M p 2 n ≅ ⨁ η   m η Ï€ η L_(phi)^(2)(Mp_(2n))~=bigoplus_(eta)m_(eta)pi_(eta)L_{\phi}^{2}\left(\mathrm{Mp}_{2 n}\right) \cong \bigoplus_{\eta} m_{\eta} \pi_{\eta}LÏ•2(Mp2n)≅⨁ηmηπη
where η = v η v η = ⊗ v η v eta=ox_(v)eta_(v)\eta=\otimes_{v} \eta_{v}η=⊗vηv runs over continuous characters of S ϕ , A S Ï• , A S_(phi,A)S_{\phi, \mathbb{A}}SÏ•,A. Here π η Ï€ η pi_(eta)\pi_{\eta}πη is defined as the restricted tensor product of representations π η v Ï€ η v pi_(eta_(v))\pi_{\eta_{v}}πηv in the local L-packets
Π ϕ v ( Mp 2 n ( F v ) ) Π Ï• v Mp 2 n ⁡ F v Pi_(phi_(v))(Mp_(2n)(F_(v)))\Pi_{\phi_{v}}\left(\operatorname{Mp}_{2 n}\left(F_{v}\right)\right)Πϕv(Mp2n⁡(Fv))
associated to ϕ v Ï• v phi_(v)\phi_{v}Ï•v (depending on ψ v ψ v psi_(v)\psi_{v}ψv ), which consist of irreducible genuine representations of Mp 2 n ( F v ) Mp 2 n ⁡ F v Mp_(2n)(F_(v))\operatorname{Mp}_{2 n}\left(F_{v}\right)Mp2n⁡(Fv) indexed by characters of S ϕ v S Ï• v S_(phi_(v))S_{\phi_{v}}SÏ•v. Also, if we define a character ε ϕ ε Ï• epsi_(phi)\varepsilon_{\phi}εϕ of ϕ â„‘ Ï• â„‘_(phi)\Im_{\phi}â„‘Ï• by
ε ϕ ( a i ) = ε ( 1 / 2 , ϕ i ) ε Ï• a i = ε 1 / 2 , Ï• i epsi_(phi)(a_(i))=epsi(1//2,phi_(i))\varepsilon_{\phi}\left(a_{i}\right)=\varepsilon\left(1 / 2, \phi_{i}\right)εϕ(ai)=ε(1/2,Ï•i)
where ε ( s , ϕ i ) ε s , Ï• i epsi(s,phi_(i))\varepsilon\left(s, \phi_{i}\right)ε(s,Ï•i) is the standard ε ε epsi\varepsilonε-function of ϕ i Ï• i phi_(i)\phi_{i}Ï•i, then m η m η m_(eta)m_{\eta}mη is given by
m η = { 1 if η Δ = ε ϕ 0 otherwise m η = 1  if  η ∘ Δ = ε Ï• 0  otherwise  m_(eta)={[1," if "eta@Delta=epsi_(phi)],[0," otherwise "]:}m_{\eta}= \begin{cases}1 & \text { if } \eta \circ \Delta=\varepsilon_{\phi} \\ 0 & \text { otherwise }\end{cases}mη={1 if η∘Δ=εϕ0 otherwise 
Remark 3.3. In fact, we gave another proof of the result of Waldspurger for M p 2 M p 2 Mp_(2)\mathrm{Mp}_{2}Mp2 [53, 56], noting that an irreducible cuspidal automorphic representation of G L 2 ( A ) G L 2 ( A ) GL_(2)(A)\mathrm{GL}_{2}(\mathbb{A})GL2(A) is symplectic if and only if its central character is trivial.
Remark 3.4. If we denote by S O 2 n + 1 S O 2 n + 1 SO_(2n+1)\mathrm{SO}_{2 n+1}SO2n+1 the split odd special orthogonal group of rank n n nnn over F F FFF and by L disc 2 ( S O 2 n + 1 ) L disc  2 S O 2 n + 1 L_("disc ")^(2)(SO_(2n+1))L_{\text {disc }}^{2}\left(\mathrm{SO}_{2 n+1}\right)Ldisc 2(SO2n+1) the discrete part of L 2 ( S O 2 n + 1 ( F ) S O 2 n + 1 ( A ) ) L 2 S O 2 n + 1 ( F ) ∖ S O 2 n + 1 ( A ) L^(2)(SO_(2n+1)(F)\\SO_(2n+1)(A))L^{2}\left(\mathrm{SO}_{2 n+1}(F) \backslash \mathrm{SO}_{2 n+1}(\mathbb{A})\right)L2(SO2n+1(F)∖SO2n+1(A)), then the decomposition of L disc 2 ( M p 2 n ) L disc  2 M p 2 n L_("disc ")^(2)(Mp_(2n))L_{\text {disc }}^{2}\left(\mathrm{Mp}_{2 n}\right)Ldisc 2(Mp2n) is similar to that of L disc 2 ( S O 2 n + 1 ) L disc  2 S O 2 n + 1 L_("disc ")^(2)(SO_(2n+1))L_{\text {disc }}^{2}\left(\mathrm{SO}_{2 n+1}\right)Ldisc 2(SO2n+1) given by Arthur [1], except that the condition η Δ = ε ϕ η ∘ Δ = ε Ï• eta@Delta=epsi_(phi)\eta \circ \Delta=\varepsilon_{\phi}η∘Δ=εϕ in the former has to be replaced by η Δ = 1 η ∘ Δ = 1 eta@Delta=1\eta \circ \Delta=1η∘Δ=1 in the latter.
In the proof of his result for n = 1 n = 1 n=1n=1n=1, Waldspurger used the theta lifting between M p 2 M p 2 Mp_(2)\mathrm{Mp}_{2}Mp2 and (inner forms of) P G L 2 S O 3 P G L 2 ≅ S O 3 PGL_(2)~=SO_(3)\mathrm{PGL}_{2} \cong \mathrm{SO}_{3}PGL2≅SO3. Thus, in general, it would be natural to use the theta lifting between M p 2 n M p 2 n Mp_(2n)\mathrm{Mp}_{2 n}Mp2n and (inner forms of) S O 2 n + 1 S O 2 n + 1 SO_(2n+1)\mathrm{SO}_{2 n+1}SO2n+1, and then transfer Arthur's endoscopic classification from S O 2 n + 1 S O 2 n + 1 SO_(2n+1)\mathrm{SO}_{2 n+1}SO2n+1 to M p 2 n M p 2 n Mp_(2n)\mathrm{Mp}_{2 n}Mp2n. However, there is a serious obstacle in this approach. Indeed, if π Ï€ pi\piÏ€ is an irreducible genuine cuspidal automorphic representation of M p 2 n ( A ) M p 2 n ( A ) Mp_(2n)(A)\mathrm{Mp}_{2 n}(\mathbb{A})Mp2n(A) and its standard L L LLL-function L ( s , π ) L ( s , Ï€ ) L(s,pi)L(s, \pi)L(s,Ï€) vanishes at s = 1 / 2 s = 1 / 2 s=1//2s=1 / 2s=1/2, then the theta lift of π Ï€ pi\piÏ€ to S O 2 n + 1 ( A ) S O 2 n + 1 ( A ) SO_(2n+1)(A)\mathrm{SO}_{2 n+1}(\mathbb{A})SO2n+1(A) is zero. When n = 1 n = 1 n=1n=1n=1, Waldspurger proved that the twisted standard L L LLL-function L ( s , π , χ ) L ( s , Ï€ , χ ) L(s,pi,chi)L(s, \pi, \chi)L(s,Ï€,χ) does not vanish at s = 1 / 2 s = 1 / 2 s=1//2s=1 / 2s=1/2 for some quadratic automorphic character χ χ chi\chiχ of A × A × A^(xx)\mathbb{A}^{\times}A×and could use the twisted theta lifting to establish the desired correspondence. But for general n n nnn, the existence of such a character χ χ chi\chiχ is considered extremely difficult to prove.
To circumvent this difficulty, we used the theta lifting in the so-called stable range studied by L i L i Li\mathrm{Li}Li [44]. More precisely, for any irreducible genuine representation π Ï€ pi\piÏ€ of M p 2 n ( A ) M p 2 n ( A ) Mp_(2n)(A)\mathrm{Mp}_{2 n}(\mathbb{A})Mp2n(A), we consider its (abstract) theta lift
θ a b s ( π ) = v θ ( π v ) θ a b s ( Ï€ ) = ⊗ v θ Ï€ v theta^(abs)(pi)=ox_(v)theta(pi_(v))\theta^{\mathrm{abs}}(\pi)=\otimes_{v} \theta\left(\pi_{v}\right)θabs(Ï€)=⊗vθ(Ï€v)
to S O 2 r + 1 ( A ) S O 2 r + 1 ( A ) SO_(2r+1)(A)\mathrm{SO}_{2 r+1}(\mathbb{A})SO2r+1(A) with r 2 n r ≫ 2 n r≫2nr \gg 2 nr≫2n. Then it follows from the result of L i L i Li\mathrm{Li}Li that if π Ï€ pi\piÏ€ occurs in L disc 2 ( M p 2 n ) L disc  2 M p 2 n L_("disc ")^(2)(Mp_(2n))L_{\text {disc }}^{2}\left(\mathrm{Mp}_{2 n}\right)Ldisc 2(Mp2n), then θ a b s ( π ) θ a b s ( Ï€ ) theta^(abs)(pi)\theta^{\mathrm{abs}}(\pi)θabs(Ï€) occurs in L disc 2 ( S O 2 r + 1 ) L disc  2 S O 2 r + 1 L_("disc ")^(2)(SO_(2r+1))L_{\text {disc }}^{2}\left(\mathrm{SO}_{2 r+1}\right)Ldisc 2(SO2r+1). Combining this with the analytic theory of standard L L LLL-functions, we may deduce Theorem 3.1 from Arthur's endoscopic classification for S O 2 r + 1 S O 2 r + 1 SO_(2r+1)\mathrm{SO}_{2 r+1}SO2r+1. Moreover, if π Ï€ pi\piÏ€ is an irreducible summand of the tempered part of L disc 2 ( M p 2 n ) L disc  2 M p 2 n L_("disc ")^(2)(Mp_(2n))L_{\text {disc }}^{2}\left(\mathrm{Mp}_{2 n}\right)Ldisc 2(Mp2n), then we proved that
m ( θ a b s ( π ) ) = m ( π ) m θ a b s ( Ï€ ) = m ( Ï€ ) m(theta^(abs)(pi))=m(pi)m\left(\theta^{\mathrm{abs}}(\pi)\right)=m(\pi)m(θabs(Ï€))=m(Ï€)
where m ( ) m ( â‹… ) m(*)m(\cdot)m(â‹…) denotes the multiplicity in the automorphic discrete spectrum. (We expect that this equality holds for any irreducible summand π Ï€ pi\piÏ€ of L disc 2 ( M p 2 n ) L disc  2 M p 2 n L_("disc ")^(2)(Mp_(2n))L_{\text {disc }}^{2}\left(\mathrm{Mp}_{2 n}\right)Ldisc 2(Mp2n).) Using this and describing the local theta lifting between M p 2 n M p 2 n Mp_(2n)\mathrm{Mp}_{2 n}Mp2n and S O 2 r + 1 S O 2 r + 1 SO_(2r+1)\mathrm{SO}_{2 r+1}SO2r+1 explicitly, we may deduce Theorem 3.2.
Remark 3.5. When n = 2 n = 2 n=2n=2n=2 and ϕ Ï• phi\phiÏ• is nontempered, we proved a similar decomposition of L ϕ 2 ( M p 4 ) L Ï• 2 M p 4 L_(phi)^(2)(Mp_(4))L_{\phi}^{2}\left(\mathrm{Mp}_{4}\right)LÏ•2(Mp4) in [21]. Note that π η Ï€ η pi_(eta)\pi_{\eta}πη is not necessarily irreducible and ε ϕ ε Ï• epsi_(phi)\varepsilon_{\phi}εϕ has to be modified in this case.

3.3. Local correspondence

There is a local analog of the above correspondence, called the local Shimura correspondence. For simplicity, we only consider the p p ppp-adic case and write F F FFF for a finite extension of Q p Q p Q_(p)\mathbb{Q}_{p}Qp. We are interested in the set
IrrMp 2 n ( F ) IrrMp 2 n ⁡ ( F ) IrrMp_(2n)(F)\operatorname{IrrMp}_{2 n}(F)IrrMp2n⁡(F)
of equivalence classes of irreducible genuine representations of the metaplectic group Mp 2 n ( F ) Mp 2 n ⁡ ( F ) Mp_(2n)(F)\operatorname{Mp}_{2 n}(F)Mp2n⁡(F). Recall that there are precisely two ( 2 n + 1 ) ( 2 n + 1 ) (2n+1)(2 n+1)(2n+1)-dimensional quadratic spaces V + V + V^(+)V^{+}V+ and V V − V^(-)V^{-}V−over F F FFF with trivial discriminant (up to isometry). Let SO ( V + ) SO ⁡ V + SO(V^(+))\operatorname{SO}\left(V^{+}\right)SO⁡(V+)and SO ( V ) SO ⁡ V − SO(V^(-))\operatorname{SO}\left(V^{-}\right)SO⁡(V−)denote the special orthogonal groups of V + V + V^(+)V^{+}V+and V V − V^(-)V^{-}V−, respectively. Then the local Shimura correspondence, which was established by Gan-Savin [22] in the p p ppp-adic case, says that there is a bijection (depending on a choice of a nontrivial additive character ψ ψ psi\psiψ of F F FFF )
θ : Irr Mp 2 n ( F ) IrrSO ( V + ) IrrSO ( V ) θ : Irr ⁡ Mp 2 n ⁡ ( F ) → IrrSO ⁡ V + ⊔ IrrSO ⁡ V − theta:Irr Mp_(2n)(F)rarr IrrSO(V^(+))⊔IrrSO(V^(-))\theta: \operatorname{Irr} \operatorname{Mp}_{2 n}(F) \rightarrow \operatorname{IrrSO}\left(V^{+}\right) \sqcup \operatorname{IrrSO}\left(V^{-}\right)θ:Irr⁡Mp2n⁡(F)→IrrSO⁡(V+)⊔IrrSO⁡(V−)
given by the local theta lifting. Namely, for any irreducible genuine representation π Ï€ pi\piÏ€ of Mp 2 n ( F ) , θ ( π ) Mp 2 n ⁡ ( F ) , θ ( Ï€ ) Mp_(2n)(F),theta(pi)\operatorname{Mp}_{2 n}(F), \theta(\pi)Mp2n⁡(F),θ(Ï€) is defined as the unique irreducible representation of SO ( V ε ) SO ⁡ V ε SO(V^(epsi))\operatorname{SO}\left(V^{\varepsilon}\right)SO⁡(Vε) with the unique sign ε = ± ε = ± epsi=+-\varepsilon= \pmε=± such that
Hom Mp 2 n ( F ) × SO ( V ε ) ( ω ε , π θ ( π ) ) 0 Hom Mp 2 n ⁡ ( F ) × SO ⁡ V ε ⁡ ω ε , Ï€ ⊗ θ ( Ï€ ) ≠ 0 Hom_(Mp_(2n)(F)xx SO(V^(epsi)))(omega^(epsi),pi ox theta(pi))!=0\operatorname{Hom}_{\operatorname{Mp}_{2 n}(F) \times \operatorname{SO}\left(V^{\varepsilon}\right)}\left(\omega^{\varepsilon}, \pi \otimes \theta(\pi)\right) \neq 0HomMp2n⁡(F)×SO⁡(Vε)⁡(ωε,π⊗θ(Ï€))≠0
where ω ε ω ε omega^(epsi)\omega^{\varepsilon}ωε is the Weil representation of Mp 2 n ( F ) × SO ( V ε ) Mp 2 n ⁡ ( F ) × SO ⁡ V ε Mp_(2n)(F)xx SO(V^(epsi))\operatorname{Mp}_{2 n}(F) \times \operatorname{SO}\left(V^{\varepsilon}\right)Mp2n⁡(F)×SO⁡(Vε) (depending on ψ ψ psi\psiψ ). Moreover, they proved various natural properties:
  • θ θ theta\thetaθ preserves the square-integrability,
  • θ θ theta\thetaθ preserves the temperedness,
  • θ θ theta\thetaθ is compatible with the theory of R R RRR-groups,
  • θ θ theta\thetaθ is compatible with the Langlands classification,
and used θ θ theta\thetaθ to transfer the local Langlands correspondence from SO ( V ε ) SO ⁡ V ε SO(V^(epsi))\operatorname{SO}\left(V^{\varepsilon}\right)SO⁡(Vε) to Mp 2 n ( F ) Mp 2 n ⁡ ( F ) Mp_(2n)(F)\operatorname{Mp}_{2 n}(F)Mp2n⁡(F). (In particular, this defines the local L L LLL-packets in the statement of Theorem 3.2.)
Remark 3.6. The local theta lifting has also been described for other reductive dual pairs in terms of the local Langlands correspondence. See [ 3 , 4 , 19 ] [ 3 , 4 , 19 ] [3,4,19][3,4,19][3,4,19] for recent progress.
As in Section 2.3, the local theta lifting can produce various relations between local analogs of periods. For example, we consider an irreducible genuine square-integrable representation π Ï€ pi\piÏ€ of Mp 2 n ( F ) Mp 2 n ⁡ ( F ) Mp_(2n)(F)\operatorname{Mp}_{2 n}(F)Mp2n⁡(F) and its formal degree d ( π ) d ( Ï€ ) d(pi)d(\pi)d(Ï€). Recall that d ( π ) d ( Ï€ ) d(pi)d(\pi)d(Ï€) is defined as the positive real number for which the Schur orthogonality relation
M p 2 n ( F ) π ( g ) v 1 , v 2 π ( g ) v 3 , v 4 ¯ d g = 1 d ( π ) v 1 , v 3 v 2 , v 4 ¯ ∫ M p 2 n ( F )   Ï€ ( g ) v 1 , v 2 Ï€ ( g ) v 3 , v 4 ¯ d g = 1 d ( Ï€ ) v 1 , v 3 v 2 , v 4 ¯ int_(Mp_(2n)(F))(:pi(g)v_(1),v_(2):) bar((:pi(g)v_(3),v_(4):))dg=(1)/(d(pi))(:v_(1),v_(3):) bar((:v_(2),v_(4):))\int_{\mathrm{Mp}_{2 n}(F)}\left\langle\pi(g) v_{1}, v_{2}\right\rangle \overline{\left\langle\pi(g) v_{3}, v_{4}\right\rangle} d g=\frac{1}{d(\pi)}\left\langle v_{1}, v_{3}\right\rangle \overline{\left\langle v_{2}, v_{4}\right\rangle}∫Mp2n(F)⟨π(g)v1,v2⟩⟨π(g)v3,v4⟩¯dg=1d(Ï€)⟨v1,v3⟩⟨v2,v4⟩¯
holds for all v 1 , , v 4 π v 1 , … , v 4 ∈ Ï€ v_(1),dots,v_(4)in piv_{1}, \ldots, v_{4} \in \piv1,…,v4∈π, where , ⟨ â‹… , â‹… ⟩ (:*,*:)\langle\cdot, \cdot\rangle⟨⋅,⋅⟩ is an invariant Hermitian inner product on π Ï€ pi\piÏ€. Note that d ( π ) d ( Ï€ ) d(pi)d(\pi)d(Ï€) depends on the choice of a Haar measure d g d g dgd gdg on Mp 2 n ( F ) Mp 2 n ⁡ ( F ) Mp_(2n)(F)\operatorname{Mp}_{2 n}(F)Mp2n⁡(F), but we take the measure determined by a Chevalley basis of the Lie algebra of Mp 2 n ( F ) Mp 2 n ⁡ ( F ) Mp_(2n)(F)\operatorname{Mp}_{2 n}(F)Mp2n⁡(F) and a fixed nontrivial additive character ψ ψ psi\psiψ of F F FFF. The above relation suggests that d ( π ) d ( Ï€ ) d(pi)d(\pi)d(Ï€) is a generalization of the dimension of an irreducible representation of a compact group, so that it is a fundamental invariant of a representation. Since a matrix coefficient g π ( g ) v 1 , v 2 g ↦ Ï€ ( g ) v 1 , v 2 g|->(:pi(g)v_(1),v_(2):)g \mapsto\left\langle\pi(g) v_{1}, v_{2}\right\rangleg↦⟨π(g)v1,v2⟩ is a local analog of an automorphic form, we may also interpret d ( π ) 1 d ( Ï€ ) − 1 d(pi)^(-1)d(\pi)^{-1}d(Ï€)−1 as a local period. Then we proved in [18] that
d ( θ ( π ) ) = d ( π ) d ( θ ( Ï€ ) ) = d ( Ï€ ) d(theta(pi))=d(pi)d(\theta(\pi))=d(\pi)d(θ(Ï€))=d(Ï€)
which is a local analog of the Rallis inner product formula [47], by using the doubling seesaw diagram:
Remark 3.7. Recall from Section 2.3 that some automorphic periods can be expressed in terms of special values of L L LLL-functions. Similarly, formal degrees should be expressed in terms of arithmetic invariants as follows. Let G G GGG be a connected reductive group over F F FFF. For simplicity, we assume that G G GGG is a pure inner form of a quasisplit group and the center of G G GGG is anisotropic. Let π Ï€ pi\piÏ€ be an irreducible square-integrable representation of G ( F ) G ( F ) G(F)G(F)G(F). Let d ( π ) d ( Ï€ ) d(pi)d(\pi)d(Ï€) denote the formal degree of π Ï€ pi\piÏ€ with respect to the Haar measure on G ( F ) G ( F ) G(F)G(F)G(F) determined by a Chevalley basis of the Lie algebra of the split form of G G GGG and a fixed nontrivial additive character ψ ψ psi\psiψ of F F FFF. Then the formal degree conjecture [29] says that
d ( π ) = dim η | S ϕ | | γ ( 0 , Ad ϕ , ψ ) | d ( Ï€ ) = dim ⁡ η S Ï• | γ ( 0 , Ad ∘ Ï• , ψ ) | d(pi)=(dim eta)/(|S_(phi)|)|gamma(0,Ad@phi,psi)|d(\pi)=\frac{\operatorname{dim} \eta}{\left|S_{\phi}\right|}|\gamma(0, \operatorname{Ad} \circ \phi, \psi)|d(Ï€)=dim⁡η|SÏ•||γ(0,Ad∘ϕ,ψ)|
where
  • ϕ : L F L G Ï• : L F → L G phi:L_(F)rarr^(L)G\phi: \mathscr{L}_{F} \rightarrow{ }^{L} GÏ•:LF→LG is the L L LLL-parameter (conjecturally) associated to π Ï€ pi\piÏ€,
  • ς ϕ Ï‚ Ï• Ï‚_(phi)\varsigma_{\phi}ςϕ is the component group of the centralizer of ϕ Ï• phi\phiÏ• in G ^ G ^ hat(G)\hat{G}G^,
  • η η eta\etaη is the irreducible representation of ς ϕ Ï‚ Ï• Ï‚_(phi)\varsigma_{\phi}ςϕ (conjecturally) associated to π Ï€ pi\piÏ€,
  • Ad is the adjoint representation of L G L G ^(L)G{ }^{L} GLG on its Lie algebra,
  • γ ( s γ ( s gamma(s\gamma(sγ(s, Ad ϕ , ψ ) ∘ Ï• , ψ ) @phi,psi)\circ \phi, \psi)∘ϕ,ψ) is the local γ γ gamma\gammaγ-factor given by
γ ( s , Ad ϕ , ψ ) = ε ( s , Ad ϕ , ψ ) L ( 1 s , Ad ϕ ) L ( s , Ad ϕ ) γ ( s , Ad ∘ Ï• , ψ ) = ε ( s , Ad ∘ Ï• , ψ ) L ( 1 − s , Ad ∘ Ï• ) L ( s , Ad ∘ Ï• ) gamma(s,Ad@phi,psi)=epsi(s,Ad@phi,psi)(L(1-s,Ad@phi))/(L(s,Ad@phi))\gamma(s, \operatorname{Ad} \circ \phi, \psi)=\varepsilon(s, \operatorname{Ad} \circ \phi, \psi) \frac{L(1-s, \operatorname{Ad} \circ \phi)}{L(s, \operatorname{Ad} \circ \phi)}γ(s,Ad∘ϕ,ψ)=ε(s,Ad∘ϕ,ψ)L(1−s,Ad∘ϕ)L(s,Ad∘ϕ)
In [34], we proved this conjecture for (inner forms of) S O 2 n + 1 S O 2 n + 1 SO_(2n+1)\mathrm{SO}_{2 n+1}SO2n+1 and its analog for M p 2 n M p 2 n Mp_(2n)\mathrm{Mp}_{2 n}Mp2n by using the main identity of Lapid-Mao [43] and the above relation between formal degrees.

4. GEOMETRIC REALIZATION OF THE JACOUET-LANGLANDS CORRESPONDENCE

Let π Ï€ pi\piÏ€ be an irreducible automorphic representation of G ( A ) G ( A ) G(A)G(\mathbb{A})G(A) and suppose that π Ï€ pi\piÏ€ is cohomological, so that π Ï€ pi\piÏ€ occurs in the cohomology H ( X , C ) H ∗ ( X , C ) H^(**)(X,C)H^{*}(X, \mathbb{C})H∗(X,C), where G G GGG is a connected reductive group and X X XXX is a locally symmetric space for G G GGG. Then it is natural to ask whether functorial transfers of π Ï€ pi\piÏ€ can be realized geometrically. In this section, we discuss the simplest example, i.e., the Jacquet-Langlands correspondence for G L 2 G L 2 GL_(2)\mathrm{GL}_{2}GL2 and its inner forms.
Let F F FFF be a totally real number field. Let A and A f A and  A f A^("and ")A_(f)\mathbb{A}^{\text {and }} \mathbb{A}_{f}Aand Af denote the rings of adèles and finite adèles of F F FFF, respectively. Let π Ï€ pi\piÏ€ be an irreducible cuspidal automorphic representation of G L 2 ( A ) G L 2 ( A ) GL_(2)(A)\mathrm{GL}_{2}(\mathbb{A})GL2(A) such that π v Ï€ v pi_(v)\pi_{v}Ï€v is the discrete series of weight 2 for all infinite places v v vvv of F F FFF. For simplicity, we assume that the central character of π Ï€ pi\piÏ€ is trivial, the level of π Ï€ pi\piÏ€ is square-free,
and the Hecke eigenvalues of π Ï€ pi\piÏ€ lie in Q Q Q\mathbb{Q}Q. Let B B BBB be a quaternion division algebra over F F FFF. For each place v v vvv of F F FFF, put B v = B F F v B v = B ⊗ F F v B_(v)=Box_(F)F_(v)B_{v}=B \otimes_{F} F_{v}Bv=B⊗FFv. Let V B V B V_(B)\mathcal{V}_{B}VB be the set of infinite places v v vvv of F F FFF such that B v B v B_(v)B_{v}Bv is split. Assume that V B V B ≠ ∅ V_(B)!=O/\mathcal{V}_{B} \neq \varnothingVB≠∅ and put d = | V B | d = V B d=|V_(B)|d=\left|\mathcal{V}_{B}\right|d=|VB|. We denote by X B X B X_(B)X_{B}XB the Shimura variety for B × B × B^(xx)B^{\times}B×(with respect to some neat open compact subgroup K f K f K_(f)K_{f}Kf of B × ( A f ) B × A f B^(xx)(A_(f))B^{\times}\left(\mathbb{A}_{f}\right)B×(Af) ), which is a d d ddd-dimensional smooth projective variety over the reflex field F F ′ F^(')F^{\prime}F′, so that
  • F F ′ F^(')F^{\prime}F′ is the number field contained in C C C\mathbb{C}C such that
Gal ( Q ¯ / F ) = { σ Gal ( Q ¯ / Q ) σ V B = V B } Gal ⁡ Q ¯ / F ′ = σ ∈ Gal ⁡ ( Q ¯ / Q ) ∣ σ V B = V B Gal( bar(Q)//F^('))={sigma in Gal( bar(Q)//Q)∣sigmaV_(B)=V_(B)}\operatorname{Gal}\left(\overline{\mathbb{Q}} / F^{\prime}\right)=\left\{\sigma \in \operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q}) \mid \sigma \mathcal{V}_{B}=\mathcal{V}_{B}\right\}Gal⁡(Q¯/F′)={σ∈Gal⁡(Q¯/Q)∣σVB=VB}
where Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯ denotes the algebraic closure of Q Q Q\mathbb{Q}Q in C C C\mathbb{C}C,
  • the C C C\mathbb{C}C-valued points of X B X B X_(B)X_{B}XB are given by
X B ( C ) = B × ( H ± ) d × B × ( A f ) / K f X B ( C ) = B × ∖ H ± d × B × A f / K f X_(B)(C)=B^(xx)\\(H^(+-))^(d)xxB^(xx)(A_(f))//K_(f)X_{B}(\mathbb{C})=B^{\times} \backslash\left(\mathfrak{H}^{ \pm}\right)^{d} \times B^{\times}\left(\mathbb{A}_{f}\right) / K_{f}XB(C)=B×∖(H±)d×B×(Af)/Kf
where S ± S ± S^(+-)\mathfrak{S}^{ \pm}S±is the union of the upper and lower half-planes.
Now assume that the Jacquet-Langlands transfer π B Ï€ B pi^(B)\pi^{B}Ï€B of π Ï€ pi\piÏ€ to B × ( A ) B × ( A ) B^(xx)(A)B^{\times}(\mathbb{A})B×(A) exists, which is the case if and only if π v Ï€ v pi_(v)\pi_{v}Ï€v is a discrete series for all v v vvv at which B B BBB is ramified, and that K f K f K_(f)K_{f}Kf is chosen appropriately so that dim ( π f B ) K f = 1 dim ⁡ Ï€ f B K f = 1 dim (pi_(f)^(B))^(K_(f))=1\operatorname{dim}\left(\pi_{f}^{B}\right)^{K_{f}}=1dim⁡(Ï€fB)Kf=1. Here π f B Ï€ f B pi_(f)^(B)\pi_{f}^{B}Ï€fB is the finite component of π B Ï€ B pi^(B)\pi^{B}Ï€B and ( π f B ) K f Ï€ f B K f (pi_(f)^(B))^(K_(f))\left(\pi_{f}^{B}\right)^{K_{f}}(Ï€fB)Kf is the space of K f K f K_(f)K_{f}Kf-fixed vectors in π f B Ï€ f B pi_(f)^(B)\pi_{f}^{B}Ï€fB. Then it follows from Matsushima's formula that π B Ï€ B pi_(B)\pi_{B}Ï€B occurs in the cohomology H ( X B , C ) H ∗ X B , C H^(**)(X_(B),C)H^{*}\left(X_{B}, \mathbb{C}\right)H∗(XB,C). More precisely, we consider the rational cohomology H ( X B , Q ) H ∗ X B , Q H^(**)(X_(B),Q)H^{*}\left(X_{B}, \mathbb{Q}\right)H∗(XB,Q) and its π Ï€ pi\piÏ€-isotypic component
H ( X B , Q ) π = { α H ( X B , Q ) T v α = χ π v ( T v ) α for all T v H v and almost all v } H ∗ X B , Q Ï€ = α ∈ H ∗ X B , Q ∣ T v α = χ Ï€ v T v α  for all  T v ∈ H v  and almost all  v H^(**)(X_(B),Q)_(pi)={alpha inH^(**)(X_(B),Q)∣T_(v)alpha=chi_(pi_(v))(T_(v))alpha" for all "T_(v)inH_(v)" and almost all "v}H^{*}\left(X_{B}, \mathbb{Q}\right)_{\pi}=\left\{\alpha \in H^{*}\left(X_{B}, \mathbb{Q}\right) \mid T_{v} \alpha=\chi_{\pi_{v}}\left(T_{v}\right) \alpha \text { for all } T_{v} \in \mathscr{H}_{v} \text { and almost all } v\right\}H∗(XB,Q)Ï€={α∈H∗(XB,Q)∣Tvα=χπv(Tv)α for all Tv∈Hv and almost all v}
where H v = Q [ K v B v × / K v ] H v = Q K v ∖ B v × / K v H_(v)=Q[K_(v)\\B_(v)^(xx)//K_(v)]\mathscr{H}_{v}=\mathbb{Q}\left[K_{v} \backslash B_{v}^{\times} / K_{v}\right]Hv=Q[Kv∖Bv×/Kv] is the Hecke algebra with respect to the standard maximal compact subgroup K v K v K_(v)K_{v}Kv of B v × G L 2 ( F v ) B v × ≅ G L 2 F v B_(v)^(xx)~=GL_(2)(F_(v))B_{v}^{\times} \cong \mathrm{GL}_{2}\left(F_{v}\right)Bv×≅GL2(Fv) and χ π v : H v Q χ Ï€ v : H v → Q chi_(pi_(v)):H_(v)rarrQ\chi_{\pi_{v}}: \mathscr{H}_{v} \rightarrow \mathbb{Q}χπv:Hv→Q is the character by which H v H v H_(v)\mathscr{H}_{v}Hv acts on π v K v Ï€ v K v pi_(v)^(K_(v))\pi_{v}^{K_{v}}Ï€vKv. Then H ( X B , Q ) π H ∗ X B , Q Ï€ H^(**)(X_(B),Q)_(pi)H^{*}\left(X_{B}, \mathbb{Q}\right)_{\pi}H∗(XB,Q)Ï€ is concentrated in the middle degree d d ddd and H d ( X B , Q ) π H d X B , Q Ï€ H^(d)(X_(B),Q)_(pi)H^{d}\left(X_{B}, \mathbb{Q}\right)_{\pi}Hd(XB,Q)Ï€ is a 2 d 2 d 2^(d)2^{d}2d-dimensional vector space over Q Q Q\mathbb{Q}Q. Moreover, for any prime â„“ â„“\ellâ„“, the â„“ â„“\ellâ„“-adic representation of Gal ( Q ¯ / F ) Gal ⁡ Q ¯ / F ′ Gal( bar(Q)//F^('))\operatorname{Gal}\left(\overline{\mathbb{Q}} / F^{\prime}\right)Gal⁡(Q¯/F′) on
H d ( X B , Q ) π Q Q H e t d ( X B × F Q ¯ , Q ) π H d X B , Q Ï€ ⊗ Q Q â„“ ≅ H e t d X B × F ′ Q ¯ , Q â„“ Ï€ H^(d)(X_(B),Q)_(pi)ox_(Q)Q_(â„“)~=H_(et)^(d)(X_(B)xx_(F^(')) bar(Q),Q_(â„“))_(pi)H^{d}\left(X_{B}, \mathbb{Q}\right)_{\pi} \otimes_{\mathbb{Q}} \mathbb{Q}_{\ell} \cong H_{\mathrm{et}}^{d}\left(X_{B} \times_{F^{\prime}} \overline{\mathbb{Q}}, \mathbb{Q}_{\ell}\right)_{\pi}Hd(XB,Q)π⊗QQℓ≅Hetd(XB×F′Q¯,Qâ„“)Ï€
is given by the so-called tensor induction of ρ π , ρ Ï€ , â„“ rho_(pi,â„“)\rho_{\pi, \ell}ρπ,â„“, where ρ π , ρ Ï€ , â„“ rho_(pi,â„“)\rho_{\pi, \ell}ρπ,â„“ is the 2 -dimensional â„“ â„“\ellâ„“-adic representation associated to π Ï€ pi\piÏ€. We remark that it depends only on ρ π , ρ Ï€ , â„“ rho_(pi,â„“)\rho_{\pi, \ell}ρπ,â„“ and V B V B V_(B)\mathcal{V}_{B}VB.
Suppose that we have two quaternion algebras B 1 B 1 B_(1)B_{1}B1 and B 2 B 2 B_(2)B_{2}B2 as above such that
V B 1 = V B 2 V B 1 = V B 2 V_(B_(1))=V_(B_(2))\mathcal{V}_{B_{1}}=\mathcal{V}_{B_{2}}VB1=VB2
Write X 1 = X B 1 X 1 = X B 1 X_(1)=X_(B_(1))X_{1}=X_{B_{1}}X1=XB1 and X 2 = X B 2 X 2 = X B 2 X_(2)=X_(B_(2))X_{2}=X_{B_{2}}X2=XB2 for the corresponding Shimura varieties, which are of the same dimension d = | V B 1 | = | V B 2 | d = V B 1 = V B 2 d=|V_(B_(1))|=|V_(B_(2))|d=\left|\mathcal{V}_{B_{1}}\right|=\left|\mathcal{V}_{B_{2}}\right|d=|VB1|=|VB2| over the same reflex field F F ′ F^(')F^{\prime}F′. Then we have
(1) an (abstract) isomorphism
H d ( X 1 , C ) π H d ( X 2 , C ) π H d X 1 , C Ï€ ≅ H d X 2 , C Ï€ H^(d)(X_(1),C)_(pi)~=H^(d)(X_(2),C)_(pi)H^{d}\left(X_{1}, \mathbb{C}\right)_{\pi} \cong H^{d}\left(X_{2}, \mathbb{C}\right)_{\pi}Hd(X1,C)π≅Hd(X2,C)Ï€
which preserves the Hodge decomposition,
(2) an (abstract) isomorphism
H d ( X 1 , Q ) π H d ( X 2 , Q ) π H d X 1 , Q â„“ Ï€ ≅ H d X 2 , Q â„“ Ï€ H^(d)(X_(1),Q_(â„“))_(pi)~=H^(d)(X_(2),Q_(â„“))_(pi)H^{d}\left(X_{1}, \mathbb{Q}_{\ell}\right)_{\pi} \cong H^{d}\left(X_{2}, \mathbb{Q}_{\ell}\right)_{\pi}Hd(X1,Qâ„“)π≅Hd(X2,Qâ„“)Ï€
of â„“ â„“\ellâ„“-adic representations of Gal ( Q ¯ / F ) Gal ⁡ Q ¯ / F ′ Gal( bar(Q)//F^('))\operatorname{Gal}\left(\overline{\mathbb{Q}} / F^{\prime}\right)Gal⁡(Q¯/F′) for all â„“ â„“\ellâ„“.
Conjecturally, these isomorphisms are obtained from a single isomorphism
H d ( X 1 , Q ) π H d ( X 2 , Q ) π H d X 1 , Q Ï€ ≅ H d X 2 , Q Ï€ H^(d)(X_(1),Q)_(pi)~=H^(d)(X_(2),Q)_(pi)H^{d}\left(X_{1}, \mathbb{Q}\right)_{\pi} \cong H^{d}\left(X_{2}, \mathbb{Q}\right)_{\pi}Hd(X1,Q)π≅Hd(X2,Q)Ï€
given as follows. By (2) and the Künneth formula, the space of Gal ( Q ¯ / F ) Gal ⁡ Q ¯ / F ′ Gal( bar(Q)//F^('))\operatorname{Gal}\left(\overline{\mathbb{Q}} / F^{\prime}\right)Gal⁡(Q¯/F′)-fixed vectors in
H 2 d ( X 1 × X 2 , Q ( d ) ) π π H 2 d X 1 × X 2 , Q â„“ ( d ) Ï€ ⊗ Ï€ H^(2d)(X_(1)xxX_(2),Q_(â„“)(d))_(pi ox pi)H^{2 d}\left(X_{1} \times X_{2}, \mathbb{Q}_{\ell}(d)\right)_{\pi \otimes \pi}H2d(X1×X2,Qâ„“(d))π⊗π
is nonzero. Hence the Tate conjecture predicts the existence of an algebraic cycle Z C H d ( X 1 × X 2 ) Z ∈ C H d X 1 × X 2 Z inCH^(d)(X_(1)xxX_(2))Z \in \mathrm{CH}^{d}\left(X_{1} \times X_{2}\right)Z∈CHd(X1×X2) which realizes (1) and (2). Namely, let p 1 p 1 p_(1)p_{1}p1 and p 2 p 2 p_(2)p_{2}p2 be the two projections
and consider the following map:
H d ( X 1 , Q ) p 1 H d ( X 1 × X 2 , Q ) [ Z ] H 3 d ( X 1 × X 2 , Q ( d ) ) p 2 H d ( X 2 , Q ) . H d X 1 , Q → p 1 ∗ H d X 1 × X 2 , Q ↓ ∪ [ Z ] H 3 d X 1 × X 2 , Q ( d ) → p 2 ∗ H d X 2 , Q . {:[H^(d)(X_(1),Q)rarr"p_(1)^(**)"H^(d)(X_(1)xxX_(2),Q)],[ darr uu[Z]],[H^(3d)(X_(1)xxX_(2),Q(d))rarr"p_(2**)"H^(d)(X_(2),Q).]:}\begin{aligned} H^{d}\left(X_{1}, \mathbb{Q}\right) \xrightarrow{p_{1}^{*}} & H^{d}\left(X_{1} \times X_{2}, \mathbb{Q}\right) \\ & \downarrow \cup[Z] \\ & H^{3 d}\left(X_{1} \times X_{2}, \mathbb{Q}(d)\right) \xrightarrow{p_{2 *}} H^{d}\left(X_{2}, \mathbb{Q}\right) . \end{aligned}Hd(X1,Q)→p1∗Hd(X1×X2,Q)↓∪[Z]H3d(X1×X2,Q(d))→p2∗Hd(X2,Q).
Then it induces an isomorphism
ι Z : H d ( X 1 , Q ) π H d ( X 2 , Q ) π ι Z : H d X 1 , Q Ï€ → H d X 2 , Q Ï€ iota_(Z):H^(d)(X_(1),Q)_(pi)rarrH^(d)(X_(2),Q)_(pi)\iota_{Z}: H^{d}\left(X_{1}, \mathbb{Q}\right)_{\pi} \rightarrow H^{d}\left(X_{2}, \mathbb{Q}\right)_{\pi}ιZ:Hd(X1,Q)π→Hd(X2,Q)Ï€
such that
  • ι Z id C ι Z ⊗ id C iota_(Z)oxid_(C)\iota_{Z} \otimes \operatorname{id}_{\mathbb{C}}ιZ⊗idC preserves the Hodge decomposition,
  • ι Z id Q ι Z ⊗ id Q â„“ iota_(Z)oxid_(Q_(â„“))\iota_{Z} \otimes \operatorname{id}_{\mathbb{Q}_{\ell}}ιZ⊗idQâ„“ is Gal ( Q ¯ / F ) Gal ⁡ Q ¯ / F ′ Gal( bar(Q)//F^('))\operatorname{Gal}\left(\overline{\mathbb{Q}} / F^{\prime}\right)Gal⁡(Q¯/F′)-equivariant for all â„“ â„“\ellâ„“.
When d = 1 d = 1 d=1d=1d=1, the existence of Z Z ZZZ in fact follows from the result of Faltings [16]. But for general d d ddd, this remains an open problem. On the other hand, noting that the Hodge conjecture reduces it to finding a Hodge cycle on X 1 × X 2 X 1 × X 2 X_(1)xxX_(2)X_{1} \times X_{2}X1×X2, i.e., an element in
H 2 d ( X 1 × X 2 , Q ) H d , d ( X 1 × X 2 ) H 2 d X 1 × X 2 , Q ∩ H d , d X 1 × X 2 H^(2d)(X_(1)xxX_(2),Q)nnH^(d,d)(X_(1)xxX_(2))H^{2 d}\left(X_{1} \times X_{2}, \mathbb{Q}\right) \cap H^{d, d}\left(X_{1} \times X_{2}\right)H2d(X1×X2,Q)∩Hd,d(X1×X2)
we gave the following evidence.
Theorem 4.1 ([35]). Assume that B 1 B 1 B_(1)B_{1}B1 and B 2 B 2 B_(2)B_{2}B2 are ramified at some infinite place v v vvv of F F FFF. Then there exists a Hodge cycle ξ ξ xi\xiξ on X 1 × X 2 X 1 × X 2 X_(1)xxX_(2)X_{1} \times X_{2}X1×X2 which induces an isomorphism
ι ξ : H d ( X 1 , Q ) π H d ( X 2 , Q ) π ι ξ : H d X 1 , Q Ï€ → H d X 2 , Q Ï€ iota_(xi):H^(d)(X_(1),Q)_(pi)rarrH^(d)(X_(2),Q)_(pi)\iota_{\xi}: H^{d}\left(X_{1}, \mathbb{Q}\right)_{\pi} \rightarrow H^{d}\left(X_{2}, \mathbb{Q}\right)_{\pi}ιξ:Hd(X1,Q)π→Hd(X2,Q)Ï€
such that
  • ι ξ i d C ι ξ ⊗ i d C iota xi oxid_(C)\iota \xi \otimes \mathrm{id}_{\mathbb{C}}ιξ⊗idC preserves the Hodge decomposition,
  • ι ξ id Q ι ξ ⊗ id Q â„“ iota_(xi)oxid_(Q_(â„“))\iota_{\xi} \otimes \operatorname{id}_{\mathbb{Q}_{\ell}}ιξ⊗idQâ„“ is Gal ( Q ¯ / F ) Gal ⁡ Q ¯ / F ′ Gal( bar(Q)//F^('))\operatorname{Gal}\left(\overline{\mathbb{Q}} / F^{\prime}\right)Gal⁡(Q¯/F′)-equivariant for all â„“ â„“\ellâ„“.
Our proof proceeds as follows. First, we choose an ambient variety X X XXX equipped with an embedding j : X 1 × X 2 X j : X 1 × X 2 ↪ X j:X_(1)xxX_(2)↪Xj: X_{1} \times X_{2} \hookrightarrow Xj:X1×X2↪X. Then we construct a class Ξ H d , d ( X ) Ξ ∈ H d , d ( X ) Xi inH^(d,d)(X)\Xi \in H^{d, d}(X)Ξ∈Hd,d(X) such that the ( π π ) ( Ï€ ⊗ Ï€ ) (pi ox pi)(\pi \otimes \pi)(π⊗π)-isotypic component ( j Ξ ) π π j ∗ Ξ Ï€ ⊗ Ï€ (j^(**)Xi)_(pi ox pi)\left(j^{*} \Xi\right)_{\pi \otimes \pi}(j∗Ξ)π⊗π of the pullback j Ξ H d , d ( X 1 × X 2 ) j ∗ Ξ ∈ H d , d X 1 × X 2 j^(**)Xi inH^(d,d)(X_(1)xxX_(2))j^{*} \Xi \in H^{d, d}\left(X_{1} \times X_{2}\right)j∗Ξ∈Hd,d(X1×X2) is nonzero. Finally, we modify Ξ Îž Xi\XiΞ in such a way that Ξ Îž Xi\XiΞ lies in H 2 d ( X , Q ) H 2 d ( X , Q ) H^(2d)(X,Q)H^{2 d}(X, \mathbb{Q})H2d(X,Q) and ξ = ( j Ξ ) π π ξ = j ∗ Ξ Ï€ ⊗ Ï€ xi=(j^(**)Xi)_(pi ox pi)\xi=\left(j^{*} \Xi\right)_{\pi \otimes \pi}ξ=(j∗Ξ)π⊗π is the desired Hodge cycle.
More precisely, fix a totally imaginary quadratic extension E E EEE of F F FFF which embeds into B 1 B 1 B_(1)B_{1}B1 and B 2 B 2 B_(2)B_{2}B2. For i = 1 , 2 i = 1 , 2 i=1,2i=1,2i=1,2, let V i = B i V i = B i V_(i)=B_(i)\mathbf{V}_{i}=B_{i}Vi=Bi be the 2-dimensional Hermitian space over E E EEE such that
G U ( V i ) = ( B i × × E × ) / F × G U V i = B i × × E × / F × GU(V_(i))=(B_(i)^(xx)xxE^(xx))//F^(xx)\mathrm{GU}\left(\mathbf{V}_{i}\right)=\left(B_{i}^{\times} \times E^{\times}\right) / F^{\times}GU(Vi)=(Bi××E×)/F×
Then we may replace X i X i X_(i)X_{i}Xi by the Shimura variety for G U ( V i ) G U V i GU(V_(i))\mathrm{GU}\left(\mathbf{V}_{i}\right)GU(Vi). Consider the 4-dimensional Hermitian space V = V 1 V 2 V = V 1 ⊕ V 2 V=V_(1)o+V_(2)\mathbf{V}=\mathbf{V}_{1} \oplus \mathbf{V}_{2}V=V1⊕V2 over E E EEE and put G = G U ( V ) G = G U ( V ) G=GU(V)\mathbf{G}=\mathrm{GU}(\mathbf{V})G=GU(V). Note that if we write v 1 , , v d v 1 , … , v d v_(1),dots,v_(d)v_{1}, \ldots, v_{d}v1,…,vd (resp. v d + 1 , , v [ F : Q ] v d + 1 , … , v [ F : Q ] v_(d+1),dots,v_([F:Q])v_{d+1}, \ldots, v_{[F: \mathbb{Q}]}vd+1,…,v[F:Q] ) for the infinite places of F F FFF at which B 1 B 1 B_(1)B_{1}B1 and B 2 B 2 B_(2)B_{2}B2 are split (resp. ramified), then we have
G ( F v i ) = { GU ( 2 , 2 ) if i d GU ( 4 ) if i > d G F v i = GU ⁡ ( 2 , 2 )  if  i ≤ d GU ⁡ ( 4 )  if  i > d G(F_(v_(i)))={[GU(2","2)," if "i <= d],[GU(4)," if "i > d]:}\mathbf{G}\left(F_{v_{i}}\right)= \begin{cases}\operatorname{GU}(2,2) & \text { if } i \leq d \\ \operatorname{GU}(4) & \text { if } i>d\end{cases}G(Fvi)={GU⁡(2,2) if i≤dGU⁡(4) if i>d
Put G = G ( F Q R ) G ∞ = G F ⊗ Q R G_(oo)=G(Fox_(Q)R)\mathbf{G}_{\infty}=\mathbf{G}\left(F \otimes_{\mathbb{Q}} \mathbb{R}\right)G∞=G(F⊗QR) and let g g ∞ g_(oo)g_{\infty}g∞ denote the complexified Lie algebra of G G ∞ G_(oo)\mathbf{G}_{\infty}G∞. Let K K ∞ K_(oo)\mathbf{K}_{\infty}K∞ be the standard maximal connected compact modulo center subgroup of G G ∞ G_(oo)\mathbf{G}_{\infty}G∞. Let X X XXX be the Shimura variety for G G G\mathbf{G}G (with respect to some neat open compact subgroup K f K f K_(f)\mathbf{K}_{f}Kf of G ( A f ) G A f G(A_(f))\mathbf{G}\left(\mathbb{A}_{f}\right)G(Af) ), which is equipped with the embedding j : X 1 × X 2 X j : X 1 × X 2 ↪ X j:X_(1)xxX_(2)↪Xj: X_{1} \times X_{2} \hookrightarrow Xj:X1×X2↪X induced by the natural embedding
G ( U ( V 1 ) × U ( V 2 ) ) G U ( V ) G U V 1 × U V 2 ↪ G U ( V ) G(U(V_(1))xxU(V_(2)))↪GU(V)\mathrm{G}\left(\mathrm{U}\left(\mathbf{V}_{1}\right) \times \mathrm{U}\left(\mathbf{V}_{2}\right)\right) \hookrightarrow \mathrm{GU}(\mathbf{V})G(U(V1)×U(V2))↪GU(V)
where the left-hand side is the subgroup of GU ( V 1 ) × G U ( V 2 ) GU ⁡ V 1 × G U V 2 GU(V_(1))xxGU(V_(2))\operatorname{GU}\left(\mathbf{V}_{1}\right) \times \mathrm{GU}\left(\mathbf{V}_{2}\right)GU⁡(V1)×GU(V2) which consists of elements with the same similitude factor. Then Matsushima's formula says that
H ( X , C ) σ m ( σ ) H ( g , K ; σ ) σ f K f H ∗ ( X , C ) ≅ ⨁ σ   m ( σ ) H ∗ g ∞ , K ∞ ; σ ∞ ⊗ σ f K f H^(**)(X,C)~=bigoplus_(sigma)m(sigma)H^(**)(g_(oo),K_(oo);sigma_(oo))oxsigma_(f)^(K_(f))H^{*}(X, \mathbb{C}) \cong \bigoplus_{\sigma} m(\sigma) H^{*}\left(\mathfrak{g}_{\infty}, \mathbf{K}_{\infty} ; \sigma_{\infty}\right) \otimes \sigma_{f}^{\mathbf{K}_{f}}H∗(X,C)≅⨁σm(σ)H∗(g∞,K∞;σ∞)⊗σfKf
where
  • σ σ sigma\sigmaσ runs over irreducible unitary representations of G ( A ) G ( A ) G(A)\mathbf{G}(\mathbb{A})G(A),
  • σ σ ∞ sigma_(oo)\sigma_{\infty}σ∞ and σ f σ f sigma_(f)\sigma_{f}σf are the infinite and finite components of σ σ sigma\sigmaσ, respectively,
  • m ( σ ) m ( σ ) m(sigma)m(\sigma)m(σ) is the multiplicity of σ σ sigma\sigmaσ in the automorphic discrete spectrum of G G G\mathbf{G}G,
  • H ( g , K ; σ ) H ∗ g ∞ , K ∞ ; σ ∞ H^(**)(g_(oo),K_(oo);sigma_(oo))H^{*}\left(g_{\infty}, \mathbf{K}_{\infty} ; \sigma_{\infty}\right)H∗(g∞,K∞;σ∞) is the relative Lie algebra cohomology,
  • σ f K f σ f K f sigma_(f)^(K_(f))\sigma_{f}^{\mathbf{K}_{f}}σfKf is the space of K f K f K_(f)\mathbf{K}_{f}Kf-fixed vectors in σ f σ f sigma_(f)\sigma_{f}σf.
Hence, to construct a class Ξ Îž Xi\XiΞ as above, we need to find an irreducible automorphic representation σ σ sigma\sigmaσ of G ( A ) G ( A ) G(A)\mathbf{G}(\mathbb{A})G(A) which satisfies the following properties:
(a) To achieve the condition Ξ H d , d ( X ) Ξ ∈ H d , d ( X ) Xi inH^(d,d)(X)\Xi \in H^{d, d}(X)Ξ∈Hd,d(X), we require that
H d , d ( g , K ; σ ) 0 H d , d g ∞ , K ∞ ; σ ∞ ≠ 0 H^(d,d)(g_(oo),K_(oo);sigma_(oo))!=0H^{d, d}\left(\mathrm{~g}_{\infty}, \mathbf{K}_{\infty} ; \sigma_{\infty}\right) \neq 0Hd,d( g∞,K∞;σ∞)≠0
If this is the case, then it follows from the result of Vogan-Zuckerman [52] that σ v i σ v i sigma_(v_(i))\sigma_{v_{i}}σvi (restricted to U ( V ) ( F v i ) ) U ( V ) F v i {:U(V)(F_(v_(i))))\left.\mathrm{U}(\mathbf{V})\left(F_{v_{i}}\right)\right)U(V)(Fvi)) is equal to
{ 1 or A q if i d 1 if i > d 1  or  A q  if  i ≤ d 1  if  i > d {[1" or "A_(q)," if "i <= d],[1," if "i > d]:}\begin{cases}\mathbf{1} \text { or } A_{\mathrm{q}} & \text { if } i \leq d \\ \mathbf{1} & \text { if } i>d\end{cases}{1 or Aq if i≤d1 if i>d
Here 1 1 1\mathbf{1}1 denotes the trivial representation and A q A q A_(q)A_{\mathfrak{q}}Aq is the cohomological representation of U ( 2 , 2 ) U ( 2 , 2 ) U(2,2)\mathrm{U}(2,2)U(2,2) associated to the θ θ theta\thetaθ-stable parabolic subalgebra q with Levi component u ( 1 , 1 ) u ( 1 , 1 ) u ( 1 , 1 ) ⊕ u ( 1 , 1 ) u(1,1)o+u(1,1)\mathfrak{u}(1,1) \oplus \mathfrak{u}(1,1)u(1,1)⊕u(1,1). We further require that σ v i = A q σ v i = A q sigma_(v_(i))=A_(q)\sigma_{v_{i}}=A_{\mathfrak{q}}σvi=Aq if i d i ≤ d i <= di \leq di≤d in order not to make σ σ sigma\sigmaσ 1-dimensional.
(b) To achieve the condition ( j Ξ ) π π 0 j ∗ Ξ Ï€ ⊗ Ï€ ≠ 0 (j^(**)Xi)_(pi ox pi)!=0\left(j^{*} \Xi\right)_{\pi \otimes \pi} \neq 0(j∗Ξ)π⊗π≠0, we require the nonvanishing of the automorphic period
σ ( π B 1 π B 2 ) ¯ C σ ⊗ Ï€ B 1 ⊗ Ï€ B 2 ¯ → C sigma ox bar((pi^(B_(1))oxpi^(B_(2))))rarrC\sigma \otimes \overline{\left(\pi^{B_{1}} \otimes \pi^{B_{2}}\right)} \rightarrow \mathbb{C}σ⊗(Ï€B1⊗πB2)¯→C
For (a), we use the following variant of the theta lifting from SL 2 SL 2 SL_(2)\operatorname{SL}_{2}SL2 to SO ( 4 , 2 ) U ( 2 , 2 ) SO ⁡ ( 4 , 2 ) ∼ U ( 2 , 2 ) SO(4,2)∼U(2,2)\operatorname{SO}(4,2) \sim \mathrm{U}(2,2)SO⁡(4,2)∼U(2,2) or S O ( 6 ) U ( 4 ) S O ( 6 ) ∼ U ( 4 ) SO(6)∼U(4)\mathrm{SO}(6) \sim \mathrm{U}(4)SO(6)∼U(4), where ∼ ∼\sim∼ denotes an isogeny. Let B B BBB be the quaternion algebra over F F FFF such that B = B 1 B 2 B = B 1 ⋅ B 2 B=B_(1)*B_(2)B=B_{1} \cdot B_{2}B=B1⋅B2 in the Brauer group, so that B B BBB is split at all infinite places of F F FFF. We may regard V = 2 V V = ∧ 2 V V=^^^(2)VV=\wedge^{2} \mathbf{V}V=∧2V as a 3-dimensional skew-Hermitian space over B B BBB such that
G U ( V ) 0 / F × G U ( V ) / E × G U ( V ) 0 / F × ≅ G U ( V ) / E × GU(V)^(0)//F^(xx)~=GU(V)//E^(xx)\mathrm{GU}(V)^{0} / F^{\times} \cong \mathrm{GU}(\mathbf{V}) / E^{\times}GU(V)0/F×≅GU(V)/E×
Let W = B W = B W=BW=BW=B be the 1-dimensional Hermitian space over B B BBB such that
G U ( W ) = B × G U ( W ) = B × GU(W)=B^(xx)\mathrm{GU}(W)=B^{\times}GU(W)=B×
Then any σ σ sigma\sigmaσ as in (a) with trivial central character is a theta lift of an irreducible cuspidal automorphic representation τ Ï„ tau\tauÏ„ of GU ( W ) ( A ) GU ⁡ ( W ) ( A ) GU(W)(A)\operatorname{GU}(W)(\mathbb{A})GU⁡(W)(A) such that τ v Ï„ v tau_(v)\tau_{v}Ï„v is the discrete series of weight 3 for all infinite places v v vvv of F F FFF. For (b), we can easily find the corresponding τ Ï„ tau\tauÏ„ by using the following seesaw diagram:
Here V = V 1 V 2 V = V 1 ⊕ V 2 V=V_(1)o+V_(2)V=V_{1} \oplus V_{2}V=V1⊕V2 is a decomposition into 1- and 2-dimensional skew-Hermitian spaces over B B BBB such that
GU ( V 1 ) 0 = E × , G U ( V 2 ) 0 = ( B 1 × × B 2 × ) / F × GU ⁡ V 1 0 = E × , G U V 2 0 = B 1 × × B 2 × / F × GU (V_(1))^(0)=E^(xx),quadGU(V_(2))^(0)=(B_(1)^(xx)xxB_(2)^(xx))//F^(xx)\operatorname{GU}\left(V_{1}\right)^{0}=E^{\times}, \quad \mathrm{GU}\left(V_{2}\right)^{0}=\left(B_{1}^{\times} \times B_{2}^{\times}\right) / F^{\times}GU⁡(V1)0=E×,GU(V2)0=(B1××B2×)/F×
and
G ( U ( V 1 ) × U ( V 2 ) ) 0 / F × G ( U ( V 1 ) × U ( V 2 ) ) / E × G U V 1 × U V 2 0 / F × ≅ G U V 1 × U V 2 / E × G(U(V_(1))xxU(V_(2)))^(0)//F^(xx)~=G(U(V_(1))xxU(V_(2)))//E^(xx)\mathrm{G}\left(\mathrm{U}\left(V_{1}\right) \times \mathrm{U}\left(V_{2}\right)\right)^{0} / F^{\times} \cong \mathrm{G}\left(\mathrm{U}\left(\mathbf{V}_{1}\right) \times \mathrm{U}\left(\mathbf{V}_{2}\right)\right) / E^{\times}G(U(V1)×U(V2))0/F×≅G(U(V1)×U(V2))/E×
For simplicity, we further assume that the Hecke eigenvalues of σ σ sigma\sigmaσ lie in Q Q Q\mathbb{Q}Q. Thus we obtain a class Ξ H d , d ( X ) Ξ ∈ H d , d ( X ) Xi inH^(d,d)(X)\Xi \in H^{d, d}(X)Ξ∈Hd,d(X) such that ξ = ( j Ξ ) π π ξ = j ∗ Ξ Ï€ ⊗ Ï€ xi=(j^(**)Xi)_(pi ox pi)\xi=\left(j^{*} \Xi\right)_{\pi \otimes \pi}ξ=(j∗Ξ)π⊗π induces an isomorphism
H d ( X 1 , C ) π H d ( X 2 , C ) π H d X 1 , C Ï€ ≅ H d X 2 , C Ï€ H^(d)(X_(1),C)_(pi)~=H^(d)(X_(2),C)_(pi)H^{d}\left(X_{1}, \mathbb{C}\right)_{\pi} \cong H^{d}\left(X_{2}, \mathbb{C}\right)_{\pi}Hd(X1,C)π≅Hd(X2,C)Ï€
To be precise, we need to use the theta lifting valued in cohomology developed by KudlaMillson [40,41]. On the other hand, we can determine the near equivalence class of σ σ sigma\sigmaσ and prove that
H 2 d ( X , C ) σ H d , d ( X ) H 2 d ( X , C ) σ ⊂ H d , d ( X ) H^(2d)(X,C)_(sigma)subH^(d,d)(X)H^{2 d}(X, \mathbb{C})_{\sigma} \subset H^{d, d}(X)H2d(X,C)σ⊂Hd,d(X)
Hence we can modify Ξ Îž Xi\XiΞ in such a way that Ξ Îž Xi\XiΞ lies in H 2 d ( X , Q ) σ H 2 d ( X , Q ) σ H^(2d)(X,Q)_(sigma)H^{2 d}(X, \mathbb{Q})_{\sigma}H2d(X,Q)σ, so that it is a Hodge cycle. Finally, it follows from the result of Kisin-Shin-Zhu [38] that
H 2 d ( X , Q ) σ Q ( d ) m H 2 d X , Q â„“ σ ≅ Q â„“ ( − d ) m H^(2d)(X,Q_(â„“))_(sigma)~=Q_(â„“)(-d)^(m)H^{2 d}\left(X, \mathbb{Q}_{\ell}\right)_{\sigma} \cong \mathbb{Q}_{\ell}(-d)^{m}H2d(X,Qâ„“)σ≅Qâ„“(−d)m
for some positive integer m m mmm, from which Theorem 4.1 follows immediately.
Remark 4.2. It is desirable to upgrade ξ ξ xi\xiξ to an absolute Hodge cycle in the sense of Deligne [14], but this remains an open problem.

ACKNOWLEDGMENTS

The author would like to thank Tamotsu Ikeda for his constant support and encouragement. The author would also like to thank Wee Teck Gan, Erez Lapid, and Kartik Prasanna for their generosity in sharing their ideas and numerous valuable discussions over the years.

FUNDING

This work was partially supported by JSPS KAKENHI Grant Number 19H01781.

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ATSUSHI ICHINO

Department of Mathematics, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan, ichino @ math.kyoto-u.ac.jp

RATIONAL

APPROXIMATIONS

OF IRRATIONAL NUMBERS

DIMITRIS KOUKOULOPOULOS

ABSTRACT

Given quantities Δ 1 , Δ 2 , 0 Δ 1 , Δ 2 , ⋯ ⩾ 0 Delta_(1),Delta_(2),cdots >= 0\Delta_{1}, \Delta_{2}, \cdots \geqslant 0Δ1,Δ2,⋯⩾0, a fundamental problem in Diophantine approximation is to understand which irrational numbers x x xxx have infinitely many reduced rational approximations a / q a / q a//qa / qa/q such that | x a / q | < Δ q | x − a / q | < Δ q |x-a//q| < Delta_(q)|x-a / q|<\Delta_{q}|x−a/q|<Δq. Depending on the choice of Δ q Δ q Delta_(q)\Delta_{q}Δq and of x x xxx, this question may be very hard. However, Duffin and Schaeffer conjectured in 1941 that if we assume a "metric" point of view, the question is governed by a simple zeroone law: writing φ φ varphi\varphiφ for Euler's totient function, we either have q = 1 φ ( q ) Δ q = ∑ q = 1 ∞   φ ( q ) Δ q = ∞ sum_(q=1)^(oo)varphi(q)Delta_(q)=oo\sum_{q=1}^{\infty} \varphi(q) \Delta_{q}=\infty∑q=1∞φ(q)Δq=∞ and then almost all irrational numbers (in the Lebesgue sense) are approximable, or q = 1 φ ( q ) Δ q < ∑ q = 1 ∞   φ ( q ) Δ q < ∞ sum_(q=1)^(oo)varphi(q)Delta_(q) < oo\sum_{q=1}^{\infty} \varphi(q) \Delta_{q}<\infty∑q=1∞φ(q)Δq<∞ and almost no irrationals are approximable. We will present the history of the Duffin-Schaeffer conjecture and the main ideas behind the recent work of Koukoulopoulos-Maynard that settled it.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 11J83; Secondary 05C40

KEYWORDS

Diophantine approximation, Duffin-Schaeffer conjecture, Metric Number Theory, compression arguments, density increment, graph theory

1. DIOPHANTINE APPROXIMATION

Let x x xxx be an irrational number. In many settings, practical and theoretical, it is important to find fractions a / q a / q a//qa / qa/q of small numerator and denominator that approximate it well. This fundamental question lies at the core of the field of Diophantine approximation.

1.1. First principles

The "high-school way" of approximating x x xxx is to use its decimal expansion. This approach produces fractions a / 10 n a / 10 n a//10^(n)a / 10^{n}a/10n such that | x a / 10 n | 10 n x − a / 10 n ≈ 10 − n |x-a//10^(n)|~~10^(-n)\left|x-a / 10^{n}\right| \approx 10^{-n}|x−a/10n|≈10−n typically. However, the error can be made much smaller if we allow more general denominators [14, THEOREM 2.1].
Theorem 1.1. If x R Q x ∈ R ∖ Q x inR\\Qx \in \mathbb{R} \backslash \mathbb{Q}x∈R∖Q, then | x a / q | < q 2 | x − a / q | < q − 2 |x-a//q| < q^(-2)|x-a / q|<q^{-2}|x−a/q|<q−2 for infinitely many pairs ( a , q ) Z × N ( a , q ) ∈ Z × N (a,q)inZxxN(a, q) \in \mathbb{Z} \times \mathbb{N}(a,q)∈Z×N.
Dirichlet (c. 1840) gave a short and clever proof of this theorem. However, his argument is nonconstructive because it uses the pigeonhole principle. This gap is filled by the theory of continued fractions (which actually precedes Dirichlet's proof).
Given any x R Q x ∈ R ∖ Q x inR\\Qx \in \mathbb{R} \backslash \mathbb{Q}x∈R∖Q, we may write x = n 0 + r 0 n 0 x = n 0 + r 0 ≈ n 0 x=n_(0)+r_(0)~~n_(0)x=n_{0}+r_{0} \approx n_{0}x=n0+r0≈n0, where n 0 = x n 0 = ⌊ x ⌋ n_(0)=|__ x __|n_{0}=\lfloor x\rfloorn0=⌊x⌋ is the integer part of x x xxx and r 0 = { x } r 0 = { x } r_(0)={x}r_{0}=\{x\}r0={x} is its fractional part. We then let n 1 = 1 / r 0 n 1 = 1 / r 0 n_(1)=|__1//r_(0)__|n_{1}=\left\lfloor 1 / r_{0}\right\rfloorn1=⌊1/r0⌋ and r 1 = { 1 / r 0 } r 1 = 1 / r 0 r_(1)={1//r_(0)}r_{1}=\left\{1 / r_{0}\right\}r1={1/r0}, so that x = n 0 + 1 / ( n 1 + r 1 ) n 0 + 1 / n 1 x = n 0 + 1 / n 1 + r 1 ≈ n 0 + 1 / n 1 x=n_(0)+1//(n_(1)+r_(1))~~n_(0)+1//n_(1)x=n_{0}+1 /\left(n_{1}+r_{1}\right) \approx n_{0}+1 / n_{1}x=n0+1/(n1+r1)≈n0+1/n1. If we repeat this process j 1 j − 1 j-1j-1j−1 more times, we find that
(1.1) x n 0 + 1 n 1 + 1 + 1 n j with n i = 1 r i 1 , r i = { 1 r i 1 } for i = 1 , , j . (1.1) x ≈ n 0 + 1 n 1 + 1 ⋯ + 1 n j  with  n i = 1 r i − 1 , r i = 1 r i − 1  for  i = 1 , … , j .  {:(1.1)x~~n_(0)+(1)/(n_(1)+(1)/(cdots+(1)/(n_(j))))quad" with "n_(i)=|__(1)/(r_(i-1))__|","r_(i)={(1)/(r_(i-1))}" for "i=1","dots","j". ":}\begin{equation*} x \approx n_{0}+\frac{1}{n_{1}+\frac{1}{\cdots+\frac{1}{n_{j}}}} \quad \text { with } n_{i}=\left\lfloor\frac{1}{r_{i-1}}\right\rfloor, r_{i}=\left\{\frac{1}{r_{i-1}}\right\} \text { for } i=1, \ldots, j \text {. } \tag{1.1} \end{equation*}(1.1)x≈n0+1n1+1⋯+1nj with ni=⌊1ri−1⌋,ri={1ri−1} for i=1,…,j. 
If we write this fraction as a j / q j a j / q j a_(j)//q_(j)a_{j} / q_{j}aj/qj in reduced form, then a calculation reveals that
(1.2) a j = n j a j 1 + a j 2 ( j 2 ) , a 1 = n 0 n 1 + 1 , a 0 = n 0 q j = n j q j 1 + q j 2 ( j 2 ) , q 1 = n 1 , q 0 = 1 (1.2) a j = n j a j − 1 + a j − 2 ( j ⩾ 2 ) , a 1 = n 0 n 1 + 1 , a 0 = n 0 q j = n j q j − 1 + q j − 2 ( j ⩾ 2 ) , q 1 = n 1 , q 0 = 1 {:[(1.2)a_(j)=n_(j)a_(j-1)+a_(j-2)quad(j >= 2)","quada_(1)=n_(0)n_(1)+1","quada_(0)=n_(0)],[q_(j)=n_(j)q_(j-1)+q_(j-2)quad(j >= 2)","quadq_(1)=n_(1)","quadq_(0)=1]:}\begin{align*} & a_{j}=n_{j} a_{j-1}+a_{j-2} \quad(j \geqslant 2), \quad a_{1}=n_{0} n_{1}+1, \quad a_{0}=n_{0} \tag{1.2}\\ & q_{j}=n_{j} q_{j-1}+q_{j-2} \quad(j \geqslant 2), \quad q_{1}=n_{1}, \quad q_{0}=1 \end{align*}(1.2)aj=njaj−1+aj−2(j⩾2),a1=n0n1+1,a0=n0qj=njqj−1+qj−2(j⩾2),q1=n1,q0=1
When j j → ∞ j rarr ooj \rightarrow \inftyj→∞, the right-hand side of (1.1), often denoted by [ n 0 ; n 1 , , n j ] n 0 ; n 1 , … , n j [n_(0);n_(1),dots,n_(j)]\left[n_{0} ; n_{1}, \ldots, n_{j}\right][n0;n1,…,nj], converges to x x xxx. The resulting representation of x x xxx is called its continued fraction expansion. The quotients a j / q j a j / q j a_(j)//q_(j)a_{j} / q_{j}aj/qj are called the convergents of this expansion and they have remarkable properties [17]. We list some of them below, with the first giving a constructive proof of Theorem 1.1.
Theorem 1.2. Assume the above set-up and notations.
(a) For each j 0 j ⩾ 0 j >= 0j \geqslant 0j⩾0, we have 1 / ( 2 q j q j + 1 ) | x a j / q j | 1 / ( q j q j + 1 ) 1 / 2 q j q j + 1 ⩽ x − a j / q j ⩽ 1 / q j q j + 1 1//(2q_(j)q_(j+1)) <= |x-a_(j)//q_(j)| <= 1//(q_(j)q_(j+1))1 /\left(2 q_{j} q_{j+1}\right) \leqslant\left|x-a_{j} / q_{j}\right| \leqslant 1 /\left(q_{j} q_{j+1}\right)1/(2qjqj+1)⩽|x−aj/qj|⩽1/(qjqj+1).
(b) For each j 0 j ⩾ 0 j >= 0j \geqslant 0j⩾0, we have | x a j / q j | = min { | x a / q | : 1 q q j } x − a j / q j = min | x − a / q | : 1 ⩽ q ⩽ q j |x-a_(j)//q_(j)|=min{|x-a//q|:1 <= q <= q_(j)}\left|x-a_{j} / q_{j}\right|=\min \left\{|x-a / q|: 1 \leqslant q \leqslant q_{j}\right\}|x−aj/qj|=min{|x−a/q|:1⩽q⩽qj}.
(c) If | x a / q | < 1 / ( 2 q 2 ) | x − a / q | < 1 / 2 q 2 |x-a//q| < 1//(2q^(2))|x-a / q|<1 /\left(2 q^{2}\right)|x−a/q|<1/(2q2) with a a aaa and q q qqq coprime, then a / q = a j / q j a / q = a j / q j a//q=a_(j)//q_(j)a / q=a_{j} / q_{j}a/q=aj/qj for some j 0 j ⩾ 0 j >= 0j \geqslant 0j⩾0.

1.2. Improving Dirichlet's approximation theorem

It is natural to ask when a qualitative improvement of Theorem 1.1 exists. Inverting this question leads us to the following definition: we say that a real number x x xxx is badly approximable if there is c = c ( x ) > 0 c = c ( x ) > 0 c=c(x) > 0c=c(x)>0c=c(x)>0 such that | x a / q | c q 2 | x − a / q | ⩾ c q − 2 |x-a//q| >= cq^(-2)|x-a / q| \geqslant c q^{-2}|x−a/q|⩾cq−2 for all ( a , q ) Z × N ( a , q ) ∈ Z × N (a,q)inZxxN(a, q) \in \mathbb{Z} \times \mathbb{N}(a,q)∈Z×N.
We can characterize approximable numbers in terms of their continued fraction expansion. Indeed, Theorem 1.2(a) and relation (1.2) imply that 1 / 4 n j + 1 q j 2 | x a j / q j | 1 1 / 4 ⩽ n j + 1 q j 2 x − a j / q j ⩽ 1 1//4 <= n_(j+1)q_(j)^(2)|x-a_(j)//q_(j)| <= 11 / 4 \leqslant n_{j+1} q_{j}^{2}\left|x-a_{j} / q_{j}\right| \leqslant 11/4⩽nj+1qj2|x−aj/qj|⩽1.
Hence, together with Theorem 1.2(c), this implies that x x xxx is badly approximable if and only if the sequence ( n j ) j = 0 n j j = 0 ∞ (n_(j))_(j=0)^(oo)\left(n_{j}\right)_{j=0}^{\infty}(nj)j=0∞ is bounded. Famously, Lagrange proved that the quadratic irrational numbers are in one-to-one correspondence with the continued fractions that are eventually periodic [17, $ 10 ] $ 10 ] $10]\$ 10]$10]. In particular, all such numbers are badly approximable.
A related concept to badly approximable numbers is the irrationality measure. For each x R x ∈ R x inRx \in \mathbb{R}x∈R, we define it to be
μ ( x ) := sup { v 0 : 0 < | x a / q | < q ν for infinitely many pairs ( a , q ) Z × N } μ ( x ) := sup v ⩾ 0 : 0 < | x − a / q | < q − ν  for infinitely many pairs  ( a , q ) ∈ Z × N mu(x):=s u p{v >= 0:0 < |x-a//q| < q^(-nu)" for infinitely many pairs "(a,q)inZxxN}\mu(x):=\sup \left\{v \geqslant 0: 0<|x-a / q|<q^{-\nu} \text { for infinitely many pairs }(a, q) \in \mathbb{Z} \times \mathbb{N}\right\}μ(x):=sup{v⩾0:0<|x−a/q|<q−ν for infinitely many pairs (a,q)∈Z×N}
Note that μ ( x ) = 1 μ ( x ) = 1 mu(x)=1\mu(x)=1μ(x)=1 if x Q x ∈ Q x inQx \in \mathbb{Q}x∈Q, whereas μ ( x ) 2 μ ( x ) ⩾ 2 mu(x) >= 2\mu(x) \geqslant 2μ(x)⩾2 if x R Q x ∈ R ∖ Q x inR\\Qx \in \mathbb{R} \backslash \mathbb{Q}x∈R∖Q by Theorem 1.1. Moreover, μ ( x ) = 2 μ ( x ) = 2 mu(x)=2\mu(x)=2μ(x)=2 if x x xxx is badly approximable. In particular, μ ( x ) = 2 μ ( x ) = 2 mu(x)=2\mu(x)=2μ(x)=2 for all quadratic irrationals x x xxx. Remarkably, Roth [25] proved that μ ( x ) = 2 μ ( x ) = 2 mu(x)=2\mu(x)=2μ(x)=2 for all algebraic irrational numbers x x xxx.
Determining the irrationality measure of various famous transcendental constants is often very hard. We do know that μ ( e ) = 2 μ ( e ) = 2 mu(e)=2\mu(e)=2μ(e)=2, where e e eee denotes Euler's constant. However, determining μ ( π ) μ ( Ï€ ) mu(pi)\mu(\pi)μ(Ï€) is a famous open problem. Towards it, Zeilberger and Zudilin [29] proved that μ ( π ) 7.10320533 μ ( Ï€ ) ⩽ 7.10320533 … mu(pi) <= 7.10320533 dots\mu(\pi) \leqslant 7.10320533 \ldotsμ(Ï€)⩽7.10320533… It is widely believed that μ ( π ) = 2 μ ( Ï€ ) = 2 mu(pi)=2\mu(\pi)=2μ(Ï€)=2.
Instead of trying to reduce the error term in Dirichlet's approximation theorem, we often require a different type of improvement: restricting the denominators q q qqq to lie in some special set S S SSS. The theory of continued fractions is of limited use for such problems because the denominators it produces satisfy rigid recursive relations (cf. (1.2)).
For rational approximation with prime or square denominators, the best results at the moment are due to Matomäki [21] and Zaharescu [28], respectively.
Theorem 1.3 (Matomäki (2009)). Let x x xxx be an irrational number and let ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0. There are infinitely many integers a a aaa and prime numbers p p ppp such that | x a / p | < p 4 / 3 + ε | x − a / p | < p − 4 / 3 + ε |x-a//p| < p^(-4//3+epsi)|x-a / p|<p^{-4 / 3+\varepsilon}|x−a/p|<p−4/3+ε.
Theorem 1.4 (Zaharescu (1995)). Let x x xxx be an irrational number and let ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0. There are infinitely many pairs ( a , q ) Z × N ( a , q ) ∈ Z × N (a,q)inZxxN(a, q) \in \mathbb{Z} \times \mathbb{N}(a,q)∈Z×N such that | x a / q 2 | < q 8 / 3 + ε x − a / q 2 < q − 8 / 3 + ε |x-a//q^(2)| < q^(-8//3+epsi)\left|x-a / q^{2}\right|<q^{-8 / 3+\varepsilon}|x−a/q2|<q−8/3+ε.
Two important open problems are to show that Theorems 1.3 and 1.4 remain true even if we replace the constants 4 / 3 4 / 3 4//34 / 34/3 and 8 / 3 8 / 3 8//38 / 38/3 by 2 and 3, respectively.

2. METRIC DIOPHANTINE APPROXIMATION

Unable to answer simple questions about the rational approximations of specific numbers, a lot of research adopted a more statistical point of view. For example, given M > 2 M > 2 M > 2M>2M>2, what proportion of real numbers have irrationality measure M ⩾ M >= M\geqslant M⩾M ? This new perspective gives rise to the theory of metric Diophantine approximation, which has a much more analytic and probabilistic flavor than the classical theory of Diophantine approximation. As we will see, the ability to ignore small pathological sets of numbers leads to a much more robust theory that provides simple and satisfactory answers to very general questions.
In order to give precise meaning to the word "proportion," we shall endow R R R\mathbb{R}R with a measure. Here, we will mainly use the Lebesgue measure (denoted by "meas").

2.1. The theorems of Khinchin and Jarník-Besicovitch

The foundational result in the field of metric Diophantine approximation was proven by Khinchin in his seminal 1924 paper [18]. It is a rather general result: given a sequence Δ 1 , Δ 2 , , 0 Δ 1 , Δ 2 , … , ⩾ 0 Delta_(1),Delta_(2),dots, >= 0\Delta_{1}, \Delta_{2}, \ldots, \geqslant 0Δ1,Δ2,…,⩾0 of "permissible margins of error," we wish to determine for which real numbers x x xxx there are infinitely many pairs ( a , q ) Z × N ( a , q ) ∈ Z × N (a,q)inZxxN(a, q) \in \mathbb{Z} \times \mathbb{N}(a,q)∈Z×N such that | x a / q | < Δ q | x − a / q | < Δ q |x-a//q| < Delta_(q)|x-a / q|<\Delta_{q}|x−a/q|<Δq. Clearly, if x x xxx has this property, so does x + 1 x + 1 x+1x+1x+1. Hence, we may focus on studying
(2.1) A := { x [ 0 , 1 ] : | x a / q | < Δ q for infinitely many pairs ( a , q ) Z × N } (2.1) A := x ∈ [ 0 , 1 ] : | x − a / q | < Δ q  for infinitely many pairs  ( a , q ) ∈ Z × N {:(2.1)A:={x in[0,1]:|x-a//q| < Delta_(q)" for infinitely many pairs "(a,q)inZxxN}:}\begin{equation*} \mathcal{A}:=\left\{x \in[0,1]:|x-a / q|<\Delta_{q} \text { for infinitely many pairs }(a, q) \in \mathbb{Z} \times \mathbb{N}\right\} \tag{2.1} \end{equation*}(2.1)A:={x∈[0,1]:|x−a/q|<Δq for infinitely many pairs (a,q)∈Z×N}
Theorem 2.1 (Khinchin (1924)). Let Δ 1 , Δ 2 , 0 Δ 1 , Δ 2 , ⋯ ⩾ 0 Delta_(1),Delta_(2),cdots >= 0\Delta_{1}, \Delta_{2}, \cdots \geqslant 0Δ1,Δ2,⋯⩾0 and let A A A\mathcal{A}A be defined as in (2.1).
(a) If q = 1 q Δ q < ∑ q = 1 ∞   q Δ q < ∞ sum_(q=1)^(oo)qDelta_(q) < oo\sum_{q=1}^{\infty} q \Delta_{q}<\infty∑q=1∞qΔq<∞, then meas ( A ) = 0 ( A ) = 0 (A)=0(\mathcal{A})=0(A)=0.
(b) If q = 1 q Δ q = ∑ q = 1 ∞   q Δ q = ∞ sum_(q=1)^(oo)qDelta_(q)=oo\sum_{q=1}^{\infty} q \Delta_{q}=\infty∑q=1∞qΔq=∞ and the sequence ( q 2 Δ q ) q = 1 q 2 Δ q q = 1 ∞ (q^(2)Delta_(q))_(q=1)^(oo)\left(q^{2} \Delta_{q}\right)_{q=1}^{\infty}(q2Δq)q=1∞ is decreasing, then meas ( A ) = 1 meas ⁡ ( A ) = 1 meas(A)=1\operatorname{meas}(\mathcal{A})=1meas⁡(A)=1.
Corollary 2.2. For almost all x R x ∈ R x inRx \in \mathbb{R}x∈R, we have | x a / q | 1 / ( q 2 log q ) | x − a / q | ⩽ 1 / q 2 log ⁡ q |x-a//q| <= 1//(q^(2)log q)|x-a / q| \leqslant 1 /\left(q^{2} \log q\right)|x−a/q|⩽1/(q2log⁡q) for infinitely many ( a , q ) Z × N ( a , q ) ∈ Z × N (a,q)inZxxN(a, q) \in \mathbb{Z} \times \mathbb{N}(a,q)∈Z×N. On the other hand, if c > 1 c > 1 c > 1c>1c>1 is fixed, then for almost every x R x ∈ R x inRx \in \mathbb{R}x∈R, the inequality | x a / q | 1 / ( q 2 log c q ) | x − a / q | ⩽ 1 / q 2 log c ⁡ q |x-a//q| <= 1//(q^(2)log^(c)q)|x-a / q| \leqslant 1 /\left(q^{2} \log ^{c} q\right)|x−a/q|⩽1/(q2logc⁡q) admits only finitely many solutions ( a , q ) Z × N ( a , q ) ∈ Z × N (a,q)inZxxN(a, q) \in \mathbb{Z} \times \mathbb{N}(a,q)∈Z×N.
In particular, Corollary 2.2 implies that the set of badly approximable numbers has null Lebesgue measure. On the other hand, it also says that almost all real numbers have irrationality measure equal to 2 . This last result is the main motivation behind the conjecture that μ ( π ) = 2 μ ( Ï€ ) = 2 mu(pi)=2\mu(\pi)=2μ(Ï€)=2 : we expect π Ï€ pi\piÏ€ to behave like a "typical" real number.
Naturally, the fact that W M := { x R : μ ( x ) M } W M := { x ∈ R : μ ( x ) ⩾ M } W_(M):={x inR:mu(x) >= M}\mathcal{W}_{M}:=\{x \in \mathbb{R}: \mu(x) \geqslant M\}WM:={x∈R:μ(x)⩾M} has null Lebesgue measure for M > 2 M > 2 M > 2M>2M>2 raises the question of determining its Hausdorff dimension (denoted by dim ( W M ) dim ⁡ W M dim(W_(M))\operatorname{dim}\left(\mathcal{W}_{M}\right)dim⁡(WM) ). Jarník [16] and Besicovitch [5] answered this question independently of each other.
Theorem 2.3 (Jarník (1928), Besicovitch (1934)). We have dim ( W M ) = 2 / M dim ⁡ W M = 2 / M dim(W_(M))=2//M\operatorname{dim}\left(\mathcal{W}_{M}\right)=2 / Mdim⁡(WM)=2/M for all M 2 M ⩾ 2 M >= 2M \geqslant 2M⩾2.

2.2. Generalizing Khinchin's theorem

Following the publication of Khinchin's theorem, research focused on weakening the assumption that q 2 Δ q q 2 Δ q ↘ q^(2)Delta_(q)↘q^{2} \Delta_{q} \searrowq2Δq↘ in part (b). Importantly, doing so would open the door to understanding rational approximations using only a restricted set of denominators. Indeed, if q 2 Δ q q 2 Δ q ↘ q^(2)Delta_(q)↘q^{2} \Delta_{q} \searrowq2Δq↘, then either Δ q > 0 Δ q > 0 Delta_(q) > 0\Delta_{q}>0Δq>0 for all q q qqq, or there is q 0 q 0 q_(0)q_{0}q0 such that Δ q = 0 Δ q = 0 Delta_(q)=0\Delta_{q}=0Δq=0 for all q q 0 q ⩾ q 0 q >= q_(0)q \geqslant q_{0}q⩾q0. The second case is trivial, since it implies A = A = ∅ A=O/\mathscr{A}=\emptysetA=∅. So, if we wish to understand Diophantine approximation with a restricted set of denominators S S SSS (which would require Δ q = 0 Δ q = 0 Delta_(q)=0\Delta_{q}=0Δq=0 for q S q ∉ S q!in Sq \notin Sq∉S ), then we must prove a version of Theorem 2.1(b) without the assumption that q 2 Δ q q 2 Δ q ↘ q^(2)Delta_(q)↘q^{2} \Delta_{q} \searrowq2Δq↘.
In order to understand better the forces at play here, it is useful to recast Khinchin's theorem in probabilistic terms. For each q q qqq, let us define the set
A q := { x [ 0 , 1 ] : there is a Z such that | x a q | < Δ q } = [ 0 , 1 ] 0 a q ( a q Δ q , a q + Δ q ) A q := x ∈ [ 0 , 1 ] :  there is  a ∈ Z  such that  x − a q < Δ q = [ 0 , 1 ] ∩ ⋃ 0 ⩽ a ⩽ q   a q − Δ q , a q + Δ q {:[A_(q):={x in[0,1]:" there is "a inZ" such that "|x-(a)/(q)| < Delta_(q)}],[=[0","1]nnuuu_(0 <= a <= q)((a)/(q)-Delta_(q),(a)/(q)+Delta_(q))]:}\begin{align*} \mathcal{A}_{q} & :=\left\{x \in[0,1]: \text { there is } a \in \mathbb{Z} \text { such that }\left|x-\frac{a}{q}\right|<\Delta_{q}\right\} \\ & =[0,1] \cap \bigcup_{0 \leqslant a \leqslant q}\left(\frac{a}{q}-\Delta_{q}, \frac{a}{q}+\Delta_{q}\right) \end{align*}Aq:={x∈[0,1]: there is a∈Z such that |x−aq|<Δq}=[0,1]∩⋃0⩽a⩽q(aq−Δq,aq+Δq)
Then A = { x [ 0 , 1 ] : x A q A = x ∈ [ 0 , 1 ] : x ∈ A q A={x in[0,1]:x inA_(q):}\mathscr{A}=\left\{x \in[0,1]: x \in \mathscr{A}_{q}\right.A={x∈[0,1]:x∈Aq infinitely often } } }\}}, which we often write as A = lim sup q A q A = lim sup q → ∞   A q A=lims u p_(q rarr oo)A_(q)\mathscr{A}=\lim \sup _{q \rightarrow \infty} \mathscr{A}_{q}A=limsupq→∞Aq. We may thus view A A A\mathscr{A}A as the event that for a number chosen uniformly at random from [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1], an infinite number of the events A 1 , A 2 , A 1 , A 2 , … A_(1),A_(2),dots\mathscr{A}_{1}, \mathscr{A}_{2}, \ldotsA1,A2,… occur. A classical result from probability theory due to Borel and Cantelli [14, LEMMAS 1.2 E 1.3] studies precisely this kind of questions.
Theorem 2.4. Let ( Ω , F , P ) ( Ω , F , P ) (Omega,F,P)(\Omega, \mathscr{F}, \mathbb{P})(Ω,F,P) be a probability space, let E 1 , E 2 , E 1 , E 2 , … E_(1),E_(2),dotsE_{1}, E_{2}, \ldotsE1,E2,… be events in that space, and let E = lim sup sin j E j E = lim sup sin j → ∞ ⁡ E j E=l i m   s u psin_(j rarr oo)E_(j)E=\limsup \sin _{j \rightarrow \infty} E_{j}E=lim supsinj→∞⁡Ej be the event that infinitely many of the E j E j E_(j)E_{j}Ej 's occur.
(a) (The first Borel-Cantelli lemma) If j = 1 P ( E j ) < ∑ j = 1 ∞   P E j < ∞ sum_(j=1)^(oo)P(E_(j)) < oo\sum_{j=1}^{\infty} \mathbb{P}\left(E_{j}\right)<\infty∑j=1∞P(Ej)<∞, then P ( E ) = 0 P ( E ) = 0 P(E)=0\mathbb{P}(E)=0P(E)=0.
(b) (The second Borel-Cantelli lemma) If j = 1 P ( E j ) = ∑ j = 1 ∞   P E j = ∞ sum_(j=1)^(oo)P(E_(j))=oo\sum_{j=1}^{\infty} \mathbb{P}\left(E_{j}\right)=\infty∑j=1∞P(Ej)=∞ and the events E 1 , E 2 , E 1 , E 2 , … E_(1),E_(2),dotsE_{1}, E_{2}, \ldotsE1,E2,… are mutually independent, then P ( E ) = 1 P ( E ) = 1 P(E)=1\mathbb{P}(E)=1P(E)=1.
Remark. Let N N NNN be the random variable that counts how many of the events E 1 , E 2 , E 1 , E 2 , … E_(1),E_(2),dotsE_{1}, E_{2}, \ldotsE1,E2,… occur. We have E [ N ] = j = 1 P ( E j ) E [ N ] = ∑ j = 1 ∞   P E j E[N]=sum_(j=1)^(oo)P(E_(j))\mathbb{E}[N]=\sum_{j=1}^{\infty} \mathbb{P}\left(E_{j}\right)E[N]=∑j=1∞P(Ej). Hence, Theorem 2.4 says that, under certain assumptions, N = N = ∞ N=ooN=\inftyN=∞ almost surely if, and only if, E [ N ] = E [ N ] = ∞ E[N]=oo\mathbb{E}[N]=\inftyE[N]=∞.
To use the above result in the setup of Khinchin's theorem, we let Ω = [ 0 , 1 ] Ω = [ 0 , 1 ] Omega=[0,1]\Omega=[0,1]Ω=[0,1] and equip it with the Lebesgue measure as its probability measure. The relevant events E j E j E_(j)E_{j}Ej are the sets A q A q A_(q)\mathscr{A}_{q}Aq. Notice that if Δ q > 1 / ( 2 q ) Δ q > 1 / ( 2 q ) Delta_(q) > 1//(2q)\Delta_{q}>1 /(2 q)Δq>1/(2q), then A q = [ 0 , 1 ] A q = [ 0 , 1 ] A_(q)=[0,1]\mathscr{A}_{q}=[0,1]Aq=[0,1], in which case A q A q A_(q)\mathscr{A}_{q}Aq occurs immediately for all x [ 0 , 1 ] x ∈ [ 0 , 1 ] x in[0,1]x \in[0,1]x∈[0,1]. In order to avoid these trivial events, we will assume from now on that
(2.3) Δ q 1 / ( 2 q ) for all q 1 , whence meas ( A q ) = 2 q Δ q (2.3) Δ q ⩽ 1 / ( 2 q )  for all  q ⩾ 1 ,  whence meas  A q = 2 q Δ q {:(2.3)Delta_(q) <= 1//(2q)quad" for all "q >= 1","quad" whence meas "(A_(q))=2qDelta_(q):}\begin{equation*} \Delta_{q} \leqslant 1 /(2 q) \quad \text { for all } q \geqslant 1, \quad \text { whence meas }\left(\mathscr{A}_{q}\right)=2 q \Delta_{q} \tag{2.3} \end{equation*}(2.3)Δq⩽1/(2q) for all q⩾1, whence meas (Aq)=2qΔq
In particular, we see that part (a) of Khinchin's theorem is a direct consequence of the first Borel-Cantelli lemma. On the other hand, the second Borel-Cantelli lemma relies crucially on the assumption that the events E j E j E_(j)E_{j}Ej are independent of each other, something that fails generically for the events A q A q A_(q)\mathscr{A}_{q}Aq. However, there are variations of the second BorelCantelli lemma, where the assumption of independence can be replaced by weaker quasiindependence conditions on the relevant events (cf. Section 3.1). From this perspective, part (b) of Khinchin's theorem can be seen as saying that the condition that the sequence ( q 2 Δ q ) q = 1 q 2 Δ q q = 1 ∞ (q^(2)Delta_(q))_(q=1)^(oo)\left(q^{2} \Delta_{q}\right)_{q=1}^{\infty}(q2Δq)q=1∞ is decreasing guarantees enough approximate independence between the events A q A q A_(q)\mathcal{A}_{q}Aq so that the conclusion of the second Borel-Cantelli lemma remains valid.
In 1941, Duffin and Schaeffer published a seminal paper [8] that studied precisely what is the right way to generalize Khinchin's theorem so that the simple zero-one law of Borel-Cantelli holds. Their starting point was the simple observation that certain choices of the quantities Δ q Δ q Delta_(q)\Delta_{q}Δq create many dependencies between the sets A q A q A_(q)\mathscr{A}_{q}Aq, thus rendering many of the denominators q q qqq redundant. Indeed, note, for example, that if Δ 3 = Δ 15 Δ 3 = Δ 15 Delta_(3)=Delta_(15)\Delta_{3}=\Delta_{15}Δ3=Δ15, then A 3 A 15 A 3 ⊆ A 15 A_(3)subeA_(15)\mathcal{A}_{3} \subseteq \mathscr{A}_{15}A3⊆A15 because each fraction with denominator 3 can also be written as a fraction with denominator 15. By exploiting this simple idea, Duffin and Schaeffer proved the following result:
Proposition 2.5. There are Δ 1 , Δ 2 , 0 Δ 1 , Δ 2 , ⋯ ⩾ 0 Delta_(1),Delta_(2),cdots >= 0\Delta_{1}, \Delta_{2}, \cdots \geqslant 0Δ1,Δ2,⋯⩾0 such that q = 1 q Δ q = ∑ q = 1 ∞   q Δ q = ∞ sum_(q=1)^(oo)qDelta_(q)=oo\sum_{q=1}^{\infty} q \Delta_{q}=\infty∑q=1∞qΔq=∞ and yet meas(A) = 0 = 0 =0=0=0. Proof. Let p 1 < p 2 < p 1 < p 2 < ⋯ p_(1) < p_(2) < cdotsp_{1}<p_{2}<\cdotsp1<p2<⋯ be the primes in increasing order, let q j = p 1 p j q j = p 1 ⋯ p j q_(j)=p_(1)cdotsp_(j)q_{j}=p_{1} \cdots p_{j}qj=p1⋯pj, and let S j = { d p j : d q j 1 } S j = d p j : d ∣ q j − 1 S_(j)={dp_(j):d∣q_(j-1)}S_{j}=\left\{d p_{j}: d \mid q_{j-1}\right\}Sj={dpj:d∣qj−1}. We then set Δ q = ( q j j log 2 j ) 1 Δ q = q j j log 2 ⁡ j − 1 Delta_(q)=(q_(j)jlog^(2)j)^(-1)\Delta_{q}=\left(q_{j} j \log ^{2} j\right)^{-1}Δq=(qjjlog2⁡j)−1 if q S j q ∈ S j q inS_(j)q \in S_{j}q∈Sj for some j 2 j ⩾ 2 j >= 2j \geqslant 2j⩾2; otherwise, we set Δ q = 0 Δ q = 0 Delta_(q)=0\Delta_{q}=0Δq=0. We claim that this choice satisfies the needed conditions.
Since A q A q j A q ⊆ A q j A_(q)subeA_(q_(j))\mathscr{A}_{q} \subseteq \mathscr{A}_{q_{j}}Aq⊆Aqj for all q S j q ∈ S j q inS_(j)q \in S_{j}q∈Sj, we have A = lim sup sin j A q j A = lim sup sin j → ∞ ⁡ A q j A=l i m   s u psin_(j rarr oo)A_(q_(j))\mathcal{A}=\limsup \sin _{j \rightarrow \infty} \mathscr{A}_{q_{j}}A=lim supsinj→∞⁡Aqj. In addition, since j = 1 q j Δ q j < ∑ j = 1 ∞   q j Δ q j < ∞ sum_(j=1)^(oo)q_(j)Delta_(q_(j)) < oo\sum_{j=1}^{\infty} q_{j} \Delta_{q_{j}}<\infty∑j=1∞qjΔqj<∞, we have meas ( lim sup j A q j ) = 0 lim sup j → ∞   A q j = 0 (lims u p_(j rarr oo)A_(q_(j)))=0\left(\lim \sup _{j \rightarrow \infty} \mathcal{A}_{q_{j}}\right)=0(limsupj→∞Aqj)=0 by Theorem 2.1(a). Hence, meas ( A ) = 0 ( A ) = 0 (A)=0(\mathcal{A})=0(A)=0, as needed. On the other hand, we have that
q 1 q Δ q = j 2 d q j 1 d p j 1 q j j log 2 j = j 2 1 j log 2 j i j 1 ( 1 + 1 p i ) ∑ q ⩾ 1   q Δ q = ∑ j ⩾ 2   ∑ d ∣ q j − 1   d p j â‹… 1 q j j log 2 ⁡ j = ∑ j ⩾ 2   1 j log 2 ⁡ j ∏ i ⩽ j − 1   1 + 1 p i sum_(q >= 1)qDelta_(q)=sum_(j >= 2)sum_(d∣q_(j-1))dp_(j)*(1)/(q_(j)jlog^(2)j)=sum_(j >= 2)(1)/(jlog^(2)j)prod_(i <= j-1)(1+(1)/(p_(i)))\sum_{q \geqslant 1} q \Delta_{q}=\sum_{j \geqslant 2} \sum_{d \mid q_{j-1}} d p_{j} \cdot \frac{1}{q_{j} j \log ^{2} j}=\sum_{j \geqslant 2} \frac{1}{j \log ^{2} j} \prod_{i \leqslant j-1}\left(1+\frac{1}{p_{i}}\right)∑q⩾1qΔq=∑j⩾2∑d∣qj−1dpjâ‹…1qjjlog2⁡j=∑j⩾21jlog2⁡j∏i⩽j−1(1+1pi)
By the Prime Number Theorem [19, THEOREM 8.1], the last product is c log j ⩾ c log ⁡ j >= c log j\geqslant c \log j⩾clog⁡j for some absolute constant c > 0 c > 0 c > 0c>0c>0. Consequently, q = 1 q Δ q = ∑ q = 1 ∞   q Δ q = ∞ sum_(q=1)^(oo)qDelta_(q)=oo\sum_{q=1}^{\infty} q \Delta_{q}=\infty∑q=1∞qΔq=∞, as claimed.
In order to avoid the above kind of counterexamples to the generalized Khinchin theorem, Duffin and Schaeffer were naturally led to consider a modified setup, where only reduced fractions are used as approximations. They thus defined
(2.4) A := { x [ 0 , 1 ] : | x a / q | < Δ q for infinitely many reduced fractions a / q } (2.4) A ∗ := x ∈ [ 0 , 1 ] : | x − a / q | < Δ q  for infinitely many reduced fractions  a / q {:(2.4)A^(**):={x in[0,1]:|x-a//q| < Delta_(q)" for infinitely many reduced fractions "a//q}:}\begin{equation*} \mathcal{A}^{*}:=\left\{x \in[0,1]:|x-a / q|<\Delta_{q} \text { for infinitely many reduced fractions } a / q\right\} \tag{2.4} \end{equation*}(2.4)A∗:={x∈[0,1]:|x−a/q|<Δq for infinitely many reduced fractions a/q}
We may write A A ∗ A^(**)\mathcal{A}^{*}A∗ as the lim sup of the sets
(2.5) A q := [ 0 , 1 ] 0 a q gcd ( a , q ) = 1 ( a q Δ q , a q + Δ q ) (2.5) A q ∗ := [ 0 , 1 ] ∩ ⋃ 0 ⩽ a ⩽ q gcd ⁡ ( a , q ) = 1   a q − Δ q , a q + Δ q {:(2.5)A_(q)^(**):=[0","1]nnuuu_({:[0 <= a <= q],[gcd(a","q)=1]:})((a)/(q)-Delta_(q),(a)/(q)+Delta_(q)):}\begin{equation*} \mathcal{A}_{q}^{*}:=[0,1] \cap \bigcup_{\substack{0 \leqslant a \leqslant q \\ \operatorname{gcd}(a, q)=1}}\left(\frac{a}{q}-\Delta_{q}, \frac{a}{q}+\Delta_{q}\right) \tag{2.5} \end{equation*}(2.5)Aq∗:=[0,1]∩⋃0⩽a⩽qgcd⁡(a,q)=1(aq−Δq,aq+Δq)
Assuming that (2.3) holds, we readily find that
meas ( A q ) = 2 φ ( q ) Δ q meas ⁡ A q ∗ = 2 φ ( q ) Δ q meas(A_(q)^(**))=2varphi(q)Delta_(q)\operatorname{meas}\left(\mathcal{A}_{q}^{*}\right)=2 \varphi(q) \Delta_{q}meas⁡(Aq∗)=2φ(q)Δq
where
φ ( q ) := # { 1 a q : gcd ( a , q ) = 1 } φ ( q ) := # { 1 ⩽ a ⩽ q : gcd ⁡ ( a , q ) = 1 } varphi(q):=#{1 <= a <= q:gcd(a,q)=1}\varphi(q):=\#\{1 \leqslant a \leqslant q: \operatorname{gcd}(a, q)=1\}φ(q):=#{1⩽a⩽q:gcd⁡(a,q)=1}
is Euler's totient function. They then conjectured that the sets A q A q ∗ A_(q)^(**)\mathscr{A}_{q}^{*}Aq∗ have enough mutual quasiindependence so that a simple zero-one law holds, as per the Borel-Cantelli lemmas.
The Duffin-Schaeffer conjecture. Let Δ 1 , Δ 2 , 0 Δ 1 , Δ 2 , ⋯ ⩾ 0 Delta_(1),Delta_(2),cdots >= 0\Delta_{1}, \Delta_{2}, \cdots \geqslant 0Δ1,Δ2,⋯⩾0 and let A A ∗ A^(**)\mathcal{A}^{*}A∗ be defined as in (2.4).
(a) If q = 1 φ ( q ) Δ q < ∑ q = 1 ∞   φ ( q ) Δ q < ∞ sum_(q=1)^(oo)varphi(q)Delta_(q) < oo\sum_{q=1}^{\infty} \varphi(q) \Delta_{q}<\infty∑q=1∞φ(q)Δq<∞, then meas ( A ) = 0 A ∗ = 0 (A^(**))=0\left(\mathscr{A}^{*}\right)=0(A∗)=0.
(b) If q = 1 φ ( q ) Δ q = ∑ q = 1 ∞   φ ( q ) Δ q = ∞ sum_(q=1)^(oo)varphi(q)Delta_(q)=oo\sum_{q=1}^{\infty} \varphi(q) \Delta_{q}=\infty∑q=1∞φ(q)Δq=∞, then meas ( A ) = 1 A ∗ = 1 (A^(**))=1\left(\mathscr{A}^{*}\right)=1(A∗)=1.
Of course, part (a) follows from Theorem 2.4(a); the main difficulty is to prove (b).
The Duffin-Schaeffer conjecture is strikingly simple and general. Nonetheless, it does not answer our original question: what is the correct generalization of Khinchin's theorem, where we may use nonreduced fractions? This gap was filled by Catlin [7].
Catlin's conjecture. Let Δ 1 , Δ 2 , 0 Δ 1 , Δ 2 , ⋯ ⩾ 0 Delta_(1),Delta_(2),cdots >= 0\Delta_{1}, \Delta_{2}, \cdots \geqslant 0Δ1,Δ2,⋯⩾0, let Δ q = sup m 1 Δ q m Δ q ′ = sup m ⩾ 1   Δ q m Delta_(q)^(')=s u p_(m >= 1)Delta_(qm)\Delta_{q}^{\prime}=\sup _{m \geqslant 1} \Delta_{q m}Δq′=supm⩾1Δqm, and let A A A\mathcal{A}A be as in (2.1).
(a) If q = 1 φ ( q ) Δ q < ∑ q = 1 ∞   φ ( q ) Δ q ′ < ∞ sum_(q=1)^(oo)varphi(q)Delta_(q)^(') < oo\sum_{q=1}^{\infty} \varphi(q) \Delta_{q}^{\prime}<\infty∑q=1∞φ(q)Δq′<∞, then meas ( A ) = 0 meas ⁡ ( A ) = 0 meas(A)=0\operatorname{meas}(\mathcal{A})=0meas⁡(A)=0.
(b) If q = 1 φ ( q ) Δ q = ∑ q = 1 ∞   φ ( q ) Δ q ′ = ∞ sum_(q=1)^(oo)varphi(q)Delta_(q)^(')=oo\sum_{q=1}^{\infty} \varphi(q) \Delta_{q}^{\prime}=\infty∑q=1∞φ(q)Δq′=∞, then meas ( A ) = 1 ( A ) = 1 (A)=1(\mathcal{A})=1(A)=1.
As Catlin noticed, his conjecture is a direct corollary of that by Duffin and Schaeffer. Indeed, let us consider the set
A = { x [ 0 , 1 ] : | x a / q | < Δ q for infinitely many reduced fractions a / q } A ′ = x ∈ [ 0 , 1 ] : | x − a / q | < Δ q ′  for infinitely many reduced fractions  a / q A^(')={x in[0,1]:|x-a//q| < Delta_(q)^(')" for infinitely many reduced fractions "a//q}\mathcal{A}^{\prime}=\left\{x \in[0,1]:|x-a / q|<\Delta_{q}^{\prime} \text { for infinitely many reduced fractions } a / q\right\}A′={x∈[0,1]:|x−a/q|<Δq′ for infinitely many reduced fractions a/q}
This is the set A A ∗ A^(**)\mathscr{A}^{*}A∗ with the quantities Δ q Δ q Delta_(q)\Delta_{q}Δq replaced by Δ q Δ q ′ Delta_(q)^(')\Delta_{q}^{\prime}Δq′, so we may apply the DuffinSchaeffer conjecture to it. In addition, it is straightforward to check that
(2.6) A Q = A Q (2.6) A ∖ Q = A ′ ∖ Q {:(2.6)A\\Q=A^(')\\Q:}\begin{equation*} \mathcal{A} \backslash \mathbb{Q}=\mathcal{A}^{\prime} \backslash \mathbb{Q} \tag{2.6} \end{equation*}(2.6)A∖Q=A′∖Q
when Δ q 0 Δ q → 0 Delta_(q)rarr0\Delta_{q} \rightarrow 0Δq→0. This settles Catlin's conjecture in this case. On the other hand, if Δ q 0 Δ q ↛ 0 Delta_(q)↛0\Delta_{q} \nrightarrow 0Δq↛0, then A = [ 0 , 1 ] A = [ 0 , 1 ] A=[0,1]\mathcal{A}=[0,1]A=[0,1] and q = 1 φ ( q ) Δ q = ∑ q = 1 ∞   φ ( q ) Δ q ′ = ∞ sum_(q=1)^(oo)varphi(q)Delta_(q)^(')=oo\sum_{q=1}^{\infty} \varphi(q) \Delta_{q}^{\prime}=\infty∑q=1∞φ(q)Δq′=∞, so that Catlin's conjecture is trivially true.
Just like in Theorem 2.3 of Jarník and Besicovitch, it would be important to also have information about the Hausdorff dimension of the sets A A A\mathcal{A}A and A A ∗ A^(**)\mathscr{A}^{*}A∗ in the case when they have null Lebesgue measure. In light of relation (2.6), it suffices to answer this question for the latter set. Beresnevich and Velani [4] proved the remarkable result that the DuffinSchaeffer conjecture implies a Hausdorff measure version of itself. This is a consequence of a much more general Mass Transfer Principle that they established, and which allows transfering information concerning the Lebesgue measure of certain lim sup sets to the Hausdorff measure of rescaled versions of them. As a corollary, they proved:
Theorem 2.6 (Beresnevich-Velani (2006)). Assume that the Duffin-Schaeffer conjecture is true. Let Δ 1 , Δ 2 , 0 Δ 1 , Δ 2 , ⋯ ⩾ 0 Delta_(1),Delta_(2),cdots >= 0\Delta_{1}, \Delta_{2}, \cdots \geqslant 0Δ1,Δ2,⋯⩾0 be such that q = 1 φ ( q ) Δ q < ∑ q = 1 ∞   φ ( q ) Δ q < ∞ sum_(q=1)^(oo)varphi(q)Delta_(q) < oo\sum_{q=1}^{\infty} \varphi(q) \Delta_{q}<\infty∑q=1∞φ(q)Δq<∞. Then the Hausdorff dimension of the set A A ∗ A^(**)\mathscr{A}^{*}A∗ defined by (2.4) equals the infimum of the set of s > 0 s > 0 s > 0s>0s>0 such that q = 1 φ ( q ) Δ q s < ∑ q = 1 ∞   φ ( q ) Δ q s < ∞ sum_(q=1)^(oo)varphi(q)Delta_(q)^(s) < oo\sum_{q=1}^{\infty} \varphi(q) \Delta_{q}^{s}<\infty∑q=1∞φ(q)Δqs<∞.

2.3. Progress towards the Duffin-Schaeffer conjecture

Since its introduction in 1941, the Duffin-Schaeffer conjectured has been the subject of intensive research activity, with various special cases proven over the years. This process came to a conclusion recently with the proof of the full conjecture [20].
Theorem 2.7 (Koukoulopoulos-Maynard (2020)). The Duffin-Schaeffer conjecture is true.
We will outline the main ideas of the proof of Theorem 2.7 in Section 3. But first we give an account of the work that preceeded it.
Notation. Given two functions f , g : X R f , g : X → R f,g:X rarrRf, g: X \rightarrow \mathbb{R}f,g:X→R, we write f ( x ) g ( x ) f ( x ) ≪ g ( x ) f(x)≪g(x)f(x) \ll g(x)f(x)≪g(x) (or f ( x ) = O ( g ( x ) f ( x ) = O ( g ( x ) f(x)=O(g(x)f(x)=O(g(x)f(x)=O(g(x) )) for all x X x ∈ X x in Xx \in Xx∈X to mean that there is a constant C C CCC such that | f ( x ) | C g ( x ) | f ( x ) | ⩽ C g ( x ) |f(x)| <= Cg(x)|f(x)| \leqslant C g(x)|f(x)|⩽Cg(x) for all x Y x ∈ Y x in Yx \in Yx∈Y.
In the same paper where they introduced their conjecture, Duffin and Schaeffer proved the first general case of it:
Theorem 2.8 (Duffin-Schaeffer (1941)). The Duffin-Schaeffer conjecture is true for all sequences ( Δ q ) q = 1 Δ q q = 1 ∞ (Delta_(q))_(q=1)^(oo)\left(\Delta_{q}\right)_{q=1}^{\infty}(Δq)q=1∞ such that
(2.7) lim sup Q q Q φ ( q ) Δ q q Q q Δ q > 0 (2.7) lim sup Q → ∞   ∑ q ⩽ Q   φ ( q ) Δ q ∑ q ⩽ Q   q Δ q > 0 {:(2.7)l i m   s u p_(Q rarr oo)(sum_(q <= Q)varphi(q)Delta_(q))/(sum_(q <= Q)qDelta_(q)) > 0:}\begin{equation*} \limsup _{Q \rightarrow \infty} \frac{\sum_{q \leqslant Q} \varphi(q) \Delta_{q}}{\sum_{q \leqslant Q} q \Delta_{q}}>0 \tag{2.7} \end{equation*}(2.7)lim supQ→∞∑q⩽Qφ(q)Δq∑q⩽QqΔq>0
To appreciate this result, we must make a few comments about condition (2.7). Note that its left-hand side is the average value of φ ( q ) / q φ ( q ) / q varphi(q)//q\varphi(q) / qφ(q)/q over q [ 1 , Q ] q ∈ [ 1 , Q ] q in[1,Q]q \in[1, Q]q∈[1,Q], where q q qqq is weighted by w q := q Δ q w q := q Δ q w_(q):=qDelta_(q)w_{q}:=q \Delta_{q}wq:=qΔq. In particular, we may restrict our attention to q q qqq with Δ q > 0 Δ q > 0 Delta_(q) > 0\Delta_{q}>0Δq>0. Now, we know
φ ( q ) q = p q ( 1 1 p ) φ ( q ) q = ∏ p ∣ q   1 − 1 p (varphi(q))/(q)=prod_(p∣q)(1-(1)/(p))\frac{\varphi(q)}{q}=\prod_{p \mid q}\left(1-\frac{1}{p}\right)φ(q)q=∏p∣q(1−1p)
In particular, φ ( q ) / q 1 φ ( q ) / q ⩽ 1 varphi(q)//q <= 1\varphi(q) / q \leqslant 1φ(q)/q⩽1, and the only way this ratio can become much smaller than 1 is if q q qqq is divisible by lots of small primes. To see this, let us begin by observing that q q qqq can have at most log q / log 2 log ⁡ q / log ⁡ 2 log q//log 2\log q / \log 2log⁡q/log⁡2 prime factors in total. Therefore,
(2.8) p q , p > log q ( 1 1 p ) ( 1 1 log q ) log q / log 2 1 5 (2.8) ∏ p ∣ q , p > log ⁡ q   1 − 1 p ⩾ 1 − 1 log ⁡ q log ⁡ q / log ⁡ 2 ⩾ 1 5 {:(2.8)prod_(p∣q,p > log q)(1-(1)/(p)) >= (1-(1)/(log q))^(log q//log 2) >= (1)/(5):}\begin{equation*} \prod_{p \mid q, p>\log q}\left(1-\frac{1}{p}\right) \geqslant\left(1-\frac{1}{\log q}\right)^{\log q / \log 2} \geqslant \frac{1}{5} \tag{2.8} \end{equation*}(2.8)∏p∣q,p>log⁡q(1−1p)⩾(1−1log⁡q)log⁡q/log⁡2⩾15
for q q qqq large enough. In addition, we have
(2.9) ( log q ) 0.01 < p log q ( 1 1 p ) ( log q ) 0.01 < p log q ( 1 1 p ) 1 200 (2.9) ∏ ( log ⁡ q ) 0.01 < p ⩽ log ⁡ q   1 − 1 p ⩾ ∏ ( log ⁡ q ) 0.01 < p ⩽ log ⁡ q   1 − 1 p ⩾ 1 200 {:(2.9)prod_({:(log q)^(0.01) < p <= log q:})(1-(1)/(p)) >= prod_((log q)^(0.01) < p <= log q)(1-(1)/(p)) >= (1)/(200):}\begin{equation*} \prod_{\substack{(\log q)^{0.01}<p \leqslant \log q}}\left(1-\frac{1}{p}\right) \geqslant \prod_{(\log q)^{0.01}<p \leqslant \log q}\left(1-\frac{1}{p}\right) \geqslant \frac{1}{200} \tag{2.9} \end{equation*}(2.9)∏(log⁡q)0.01<p⩽log⁡q(1−1p)⩾∏(log⁡q)0.01<p⩽log⁡q(1−1p)⩾1200
for q q qqq large enough by Mertens' estimate [19, THEOREM 3.4]. Already the above inequalities show that only the primes ( log q ) 0.01 ⩽ ( log ⁡ q ) 0.01 <= (log q)^(0.01)\leqslant(\log q)^{0.01}⩽(log⁡q)0.01 can affect the size of φ ( q ) / q φ ( q ) / q varphi(q)//q\varphi(q) / qφ(q)/q. But more is true: φ ( q ) / q φ ( q ) / q varphi(q)//q\varphi(q) / qφ(q)/q is small only if q q qqq is divided by many primes ( log q ) 0.01 ⩽ ( log ⁡ q ) 0.01 <= (log q)^(0.01)\leqslant(\log q)^{0.01}⩽(log⁡q)0.01. Imagine, for example, that
(2.10) # { p q : e j 1 < p e j } e j / j 2 + 1000 (2.10) # p ∣ q : e j − 1 < p ⩽ e j ⩽ e j / j 2 + 1000 {:(2.10)#{p∣q:e^(j-1) < p <= e^(j)} <= e^(j)//j^(2)+1000:}\begin{equation*} \#\left\{p \mid q: e^{j-1}<p \leqslant e^{j}\right\} \leqslant e^{j} / j^{2}+1000 \tag{2.10} \end{equation*}(2.10)#{p∣q:ej−1<p⩽ej}⩽ej/j2+1000
for j = 1 , 2 , , 1 + 0.01 log log q j = 1 , 2 , … , 1 + ⌊ 0.01 log ⁡ log ⁡ q ⌋ j=1,2,dots,1+|__0.01 log log q __|j=1,2, \ldots, 1+\lfloor 0.01 \log \log q\rfloorj=1,2,…,1+⌊0.01log⁡log⁡q⌋. We would then have
j 1 < p e j p q ( 1 1 p ) ( 1 1 e j ) e j / j 2 + 1000 = exp ( 1 / j 2 + O ( e j ) ) ∏ j − 1 < p ⩽ e j p ∣ q   1 − 1 p ⩾ 1 − 1 e j e j / j 2 + 1000 = exp ⁡ − 1 / j 2 + O e − j prod_({:[j-1 < p <= e^(j)],[p∣q]:})(1-(1)/(p)) >= (1-(1)/(e^(j)))^(e^(j)//j^(2)+1000)=exp(-1//j^(2)+O(e^(-j)))\prod_{\substack{j-1<p \leqslant e^{j} \\ p \mid q}}\left(1-\frac{1}{p}\right) \geqslant\left(1-\frac{1}{e^{j}}\right)^{e^{j} / j^{2}+1000}=\exp \left(-1 / j^{2}+O\left(e^{-j}\right)\right)∏j−1<p⩽ejp∣q(1−1p)⩾(1−1ej)ej/j2+1000=exp⁡(−1/j2+O(e−j))
Multiplying this over all j j jjj, we deduce that φ ( q ) / q c φ ( q ) / q ⩾ c varphi(q)//q >= c\varphi(q) / q \geqslant cφ(q)/q⩾c for some c > 0 c > 0 c > 0c>0c>0 independent of q q qqq.
We have thus proven that for (2.7) to fail, the main contribution to the weighted sum q Q w q ∑ q ⩽ Q   w q sum_(q <= Q)w_(q)\sum_{q \leqslant Q} w_{q}∑q⩽Qwq with w q = q Δ q w q = q Δ q w_(q)=qDelta_(q)w_{q}=q \Delta_{q}wq=qΔq must come from integers for which (2.10) fails. As a matter of fact, (2.10) must fail for lots of j j jjj 's. This is an extremely rare event if we choose q q qqq uniformly at random from [ 1 , Q ] [ 1 , Q ] [1,Q][1, Q][1,Q] (or even if we choose it uniformly at random from various "nice" subsets of [ 1 , Q ] [ 1 , Q ] [1,Q][1, Q][1,Q], such as the primes, or the values of a monic polynomial with integer coefficients).

p e j } p ⩽ e j {:p <= e^(j)}\left.p \leqslant e^{j}\right\}p⩽ej} with respect to the uniform counting measure on [ 1 , Q ] [ 1 , Q ] [1,Q][1, Q][1,Q]. We have
1 Q q Q # { p q : e j 1 < p e j } = e j 1 < p e j # { q Q : p q } Q e j 1 < p e j 1 p 1 j 1 Q ∑ q ⩽ Q   # p ∣ q : e j − 1 < p ⩽ e j = ∑ e j − 1 < p ⩽ e j   # { q ⩽ Q : p ∣ q } Q ⩽ ∑ e j − 1 < p ⩽ e j   1 p ≪ 1 j (1)/(Q)sum_(q <= Q)#{p∣q:e^(j-1) < p <= e^(j)}=sum_(e^(j-1) < p <= e^(j))(#{q <= Q:p∣q})/(Q) <= sum_(e^(j-1) < p <= e^(j))(1)/(p)≪(1)/(j)\frac{1}{Q} \sum_{q \leqslant Q} \#\left\{p \mid q: e^{j-1}<p \leqslant e^{j}\right\}=\sum_{e^{j-1}<p \leqslant e^{j}} \frac{\#\{q \leqslant Q: p \mid q\}}{Q} \leqslant \sum_{e^{j-1}<p \leqslant e^{j}} \frac{1}{p} \ll \frac{1}{j}1Q∑q⩽Q#{p∣q:ej−1<p⩽ej}=∑ej−1<p⩽ej#{q⩽Q:p∣q}Q⩽∑ej−1<p⩽ej1p≪1j
by Mertens' theorem [19, THEOREM 3.4]. This is much smaller than e j / j 2 e j / j 2 e^(j)//j^(2)e^{j} / j^{2}ej/j2, so (2.10) should fail rarely as j j → ∞ j rarr ooj \rightarrow \inftyj→∞. (For instance, we may use Markov's inequality to see this claim.)
In conclusion, Theorem 2.8 settles the Duffin-Schaeffer conjecture when Δ q Δ q Delta_(q)\Delta_{q}Δq is mainly supported on "normal" integers, without too many small prime factors. In particular, it implies a significant improvement of Theorems 1.3 and 1.4 for almost all x R x ∈ R x inRx \in \mathbb{R}x∈R.
Corollary 2.9. For almost all x R x ∈ R x inRx \in \mathbb{R}x∈R, there are infinitely many reduced fractions a / p a / p a//pa / pa/p and b / q 2 b / q 2 b//q^(2)b / q^{2}b/q2 such that p p ppp is prime, | x a / p | < p 2 | x − a / p | < p − 2 |x-a//p| < p^(-2)|x-a / p|<p^{-2}|x−a/p|<p−2 and | x b / q 2 | < q 3 x − b / q 2 < q − 3 |x-b//q^(2)| < q^(-3)\left|x-b / q^{2}\right|<q^{-3}|x−b/q2|<q−3.
The next important step towards the Duffin-Schaeffer conjecture is a remarkable zero-one law due to Gallagher [12].
Theorem 2.10 (Gallagher (1961)). If A A ∗ A^(**)\mathcal{A}^{*}A∗ is as in (2.4), then meas ( A ) { 0 , 1 } A ∗ ∈ { 0 , 1 } (A^(**))in{0,1}\left(\mathcal{A}^{*}\right) \in\{0,1\}(A∗)∈{0,1}.
Gallagher's theorem says grosso modo that either we chose the quantities Δ q Δ q Delta_(q)\Delta_{q}Δq to be "too small" and thus missed almost all real numbers, or we chose them "sufficiently large" so that almost all numbers have the desired rational approximations. The Duffin-Schaeffer conjecture is then the simplest possible criterion to decide in which case we are.
The proof of Theorem 2.10 is a clever adaptation of an ergodic-theoretic argument due to Cassels [6] in the simpler setting of nonreduced rational approximations. We give Cassel's proof and refer the interested readers to [ 12 , 14 ] [ 12 , 14 ] [12,14][12,14][12,14] for the proof of Theorem 2.10.
Theorem 2.11 (Cassels (1950)). If A A A\mathscr{A}A is as in (2.1), then meas(A) ) { 0 , 1 } ) ∈ { 0 , 1 } )in{0,1}) \in\{0,1\})∈{0,1}.
Proof. We need the following fact [14, LEMMA 2.1] that uses Lebesgue's Density Theorem: Let I 1 , I 2 , , J 1 , J 2 , I 1 , I 2 , … , J 1 , J 2 , … I_(1),I_(2),dots,J_(1),J_(2),dotsI_{1}, I_{2}, \ldots, J_{1}, J_{2}, \ldotsI1,I2,…,J1,J2,… be intervals of lengths tending to 0 , and let c > 0 c > 0 c > 0c>0c>0. For all k k kkk, suppose J k I k J k ⊆ I k J_(k)subeI_(k)J_{k} \subseteq I_{k}Jk⊆Ik and meas ( J k ) c J k ⩾ c (J_(k)) >= c\left(J_{k}\right) \geqslant c(Jk)⩾c meas ( I k ) I k (I_(k))\left(I_{k}\right)(Ik). Then meas(lim sup k I k lim sup sup k J k ) = 0 k → ∞ I k ∖ lim sup sup k → ∞   J k = 0 {:_(k rarr oo)I_(k)\\l i m   s u ps u p_(k rarr oo)J_(k))=0\left.{ }_{k \rightarrow \infty} I_{k} \backslash \limsup \sup _{k \rightarrow \infty} J_{k}\right)=0k→∞Ik∖lim supsupk→∞Jk)=0.
Now, for each r 1 r ⩾ 1 r >= 1r \geqslant 1r⩾1, let A ( r ) A ( r ) A^((r))\mathscr{A}^{(r)}A(r) be defined as in (2.1) but with Δ q / r Δ q / r Delta_(q)//r\Delta_{q} / rΔq/r in place of Δ q Δ q Delta_(q)\Delta_{q}Δq. Hence, meas ( A A ( r ) ) = 0 A ∖ A ( r ) = 0 (A\\A^((r)))=0\left(\mathcal{A} \backslash \mathcal{A}^{(r)}\right)=0(A∖A(r))=0 by the above fact. Therefore, if A ( ) := n = 1 A ( n ) A ( ∞ ) := â‹‚ n = 1 ∞   A ( n ) A^((oo)):=nnn_(n=1)^(oo)A^((n))\mathcal{A}^{(\infty)}:=\bigcap_{n=1}^{\infty} \mathcal{A}^{(n)}A(∞):=â‹‚n=1∞A(n), then meas ( A A ( ) ) = 0 A ∖ A ( ∞ ) = 0 (A\\A^((oo)))=0\left(\mathcal{A} \backslash \mathcal{A}^{(\infty)}\right)=0(A∖A(∞))=0. Now, consider the map ψ : [ 0 , 1 ] [ 0 , 1 ] ψ : [ 0 , 1 ] → [ 0 , 1 ] psi:[0,1]rarr[0,1]\psi:[0,1] \rightarrow[0,1]ψ:[0,1]→[0,1] defined by ψ ( x ) := { 2 x } ψ ( x ) := { 2 x } psi(x):={2x}\psi(x):=\{2 x\}ψ(x):={2x}, and note that ψ ( A ( ) ) A ( ) ψ A ( ∞ ) ⊆ A ( ∞ ) psi(A^((oo)))subeA^((oo))\psi\left(\mathcal{A}^{(\infty)}\right) \subseteq \mathcal{A}^{(\infty)}ψ(A(∞))⊆A(∞). In particular, 1 N n = 0 N 1 1 A ( ) ( ψ n ( x ) ) = 1 1 N ∑ n = 0 N − 1   1 A ( ∞ ) ψ n ( x ) = 1 (1)/(N)sum_(n=0)^(N-1)1_(A^((oo)))(psi^(n)(x))=1\frac{1}{N} \sum_{n=0}^{N-1} 1_{\mathcal{A}^{(\infty)}}\left(\psi^{n}(x)\right)=11N∑n=0N−11A(∞)(ψn(x))=1 for all x A ( ) x ∈ A ( ∞ ) x inA^((oo))x \in \mathcal{A}^{(\infty)}x∈A(∞) and all N N N ∈ N N inNN \in \mathbb{N}N∈N. Since ψ ψ psi\psiψ is ergodic with respect to the Lebesgue measure [26, P. 293 ₹ 305-6], Birkhoff's Ergodic Theorem [26, сH. 6, coR. 5.6] implies that meas ( A ( ) ) { 0 , 1 } A ( ∞ ) ∈ { 0 , 1 } (A^((oo)))in{0,1}\left(\mathcal{A}^{(\infty)}\right) \in\{0,1\}(A(∞))∈{0,1}.
The first significant step towards establishing the Duffin-Schaeffer conjecture for irregular sequences Δ q Δ q Delta_(q)\Delta_{q}Δq, potentially supported on integers with lots of small prime factors, was carried out by ErdÅ‘s [10] and Vaaler [27].
Theorem 2.12 (ErdÅ‘s (1970), Vaaler (1978)). The Duffin-Schaeffer conjecture is true for all sequences ( Δ q ) q = 1 Δ q q = 1 ∞ (Delta_(q))_(q=1)^(oo)\left(\Delta_{q}\right)_{q=1}^{\infty}(Δq)q=1∞ such that Δ q = O ( 1 / q 2 ) Δ q = O 1 / q 2 Delta_(q)=O(1//q^(2))\Delta_{q}=O\left(1 / q^{2}\right)Δq=O(1/q2) for all q q qqq.
This theorem is, of course, most interesting when q = 1 φ ( q ) Δ q = ∑ q = 1 ∞   φ ( q ) Δ q = ∞ sum_(q=1)^(oo)varphi(q)Delta_(q)=oo\sum_{q=1}^{\infty} \varphi(q) \Delta_{q}=\infty∑q=1∞φ(q)Δq=∞. Since Δ q = O ( 1 / q 2 ) Δ q = O 1 / q 2 Delta_(q)=O(1//q^(2))\Delta_{q}=O\left(1 / q^{2}\right)Δq=O(1/q2) and φ ( q ) q φ ( q ) ⩽ q varphi(q) <= q\varphi(q) \leqslant qφ(q)⩽q, we find q S 1 / q = ∑ q ∈ S   1 / q = ∞ sum_(q inS)1//q=oo\sum_{q \in \mathcal{S}} 1 / q=\infty∑q∈S1/q=∞ with S = { q : Δ q > 0 } S = q : Δ q > 0 S={q:Delta_(q) > 0}S=\left\{q: \Delta_{q}>0\right\}S={q:Δq>0}. In particular, S S SSS must be somewhat dense in N N N\mathbb{N}N. Therefore, Theorem 2.12 has the advantage over Theorem 2.8 that S S SSS can contain many irregular integers, and the disadvantage that it has to be quite dense.
The Duffin-Schaeffer conjecture has a natural analogue in R k R k R^(k)\mathbb{R}^{k}Rk with k 2 k ⩾ 2 k >= 2k \geqslant 2k⩾2 : given Δ 1 , Δ 2 , 0 Δ 1 , Δ 2 , ⋯ ⩾ 0 Delta_(1),Delta_(2),cdots >= 0\Delta_{1}, \Delta_{2}, \cdots \geqslant 0Δ1,Δ2,⋯⩾0, let A ( k ) A ∗ ( k ) A^(**)(k)\mathcal{A}^{*}(k)A∗(k) be the set of x = ( x 1 , , x k ) R k x → = x 1 , … , x k ∈ R k vec(x)=(x_(1),dots,x_(k))inR^(k)\vec{x}=\left(x_{1}, \ldots, x_{k}\right) \in \mathbb{R}^{k}x→=(x1,…,xk)∈Rk for which there are infinitely many k k kkk-tuples ( a 1 / q , , a k / q ) a 1 / q , … , a k / q (a_(1)//q,dots,a_(k)//q)\left(a_{1} / q, \ldots, a_{k} / q\right)(a1/q,…,ak/q) of reduced fractions with | x j a j / q | < Δ q x j − a j / q < Δ q |x_(j)-a_(j)//q| < Delta_(q)\left|x_{j}-a_{j} / q\right|<\Delta_{q}|xj−aj/q|<Δq for all j j jjj. Then A ( k ) A ∗ ( k ) A^(**)(k)\mathscr{A}^{*}(k)A∗(k) should contain almost no or almost all x R k x → ∈ R k vec(x)inR^(k)\vec{x} \in \mathbb{R}^{k}x→∈Rk, according to whether the series q = 1 ( φ ( q ) Δ q ) k ∑ q = 1 ∞   φ ( q ) Δ q k sum_(q=1)^(oo)(varphi(q)Delta_(q))^(k)\sum_{q=1}^{\infty}\left(\varphi(q) \Delta_{q}\right)^{k}∑q=1∞(φ(q)Δq)k converges or diverges. This was proven by Pollington and Vaughan [22].
Theorem 2.13 (Pollington-Vaughan (1990)). The k k kkk-dimensional Duffin-Schaeffer conjecture is true for all k 2 k ⩾ 2 k >= 2k \geqslant 2k⩾2.
Following this result, a lot of research focused on proving the Duffin-Schaeffer conjecture when the series q = 1 φ ( q ) Δ q ∑ q = 1 ∞   φ ( q ) Δ q sum_(q=1)^(oo)varphi(q)Delta_(q)\sum_{q=1}^{\infty} \varphi(q) \Delta_{q}∑q=1∞φ(q)Δq diverges fast enough (see, e.g., [14, THEOREM 3.7(III)], [3,15]). Aistleitner, Lachmann, Munsch, Technau, and Zafeiropoulos [2] proved the Duffin-
Schaeffer conjecture when q = 1 φ ( q ) Δ q / ( log q ) ε = ∑ q = 1 ∞   φ ( q ) Δ q / ( log ⁡ q ) ε = ∞ sum_(q=1)^(oo)varphi(q)Delta_(q)//(log q)^(epsi)=oo\sum_{q=1}^{\infty} \varphi(q) \Delta_{q} /(\log q)^{\varepsilon}=\infty∑q=1∞φ(q)Δq/(log⁡q)ε=∞ for some ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0. A report by Aistleitner [1], announced at the same time as [20], explains how to replace ( log q ) ε ( log ⁡ q ) ε (log q)^(epsi)(\log q)^{\varepsilon}(log⁡q)ε by ( log log q ) ε ( log ⁡ log ⁡ q ) ε (log log q)^(epsi)(\log \log q)^{\varepsilon}(log⁡log⁡q)ε.

3. THE MAIN INGREDIENTS OF THE PROOF OF THE DUFFIN-SCHAEFFER CONJECTURE

3.1. Borel-Cantelli without independence

Recall the definition of the sets A q A q ∗ A_(q)^(**)\mathcal{A}_{q}^{*}Aq∗ in (2.5). Let us assume that Δ q 1 / ( 2 q ) Δ q ⩽ 1 / ( 2 q ) Delta_(q) <= 1//(2q)\Delta_{q} \leqslant 1 /(2 q)Δq⩽1/(2q) for all q q qqq (cf. (2.3)) so that meas ( A q ) = 2 φ ( q ) Δ q [ 0 , 1 ] A q ∗ = 2 φ ( q ) Δ q ∈ [ 0 , 1 ] (A_(q)^(**))=2varphi(q)Delta_(q)in[0,1]\left(\mathscr{A}_{q}^{*}\right)=2 \varphi(q) \Delta_{q} \in[0,1](Aq∗)=2φ(q)Δq∈[0,1], and let us also suppose that q = 1 φ ( q ) Δ q = ∑ q = 1 ∞   φ ( q ) Δ q = ∞ sum_(q=1)^(oo)varphi(q)Delta_(q)=oo\sum_{q=1}^{\infty} \varphi(q) \Delta_{q}=\infty∑q=1∞φ(q)Δq=∞. The first technical difficulty we must deal with is how to prove an analogue of the second Borel-Cantelli lemma (cf. Theorem 2.4(b)) without assuming that the events A q A q ∗ A_(q)^(**)\mathcal{A}_{q}^{*}Aq∗ are independent. We follow an idea due to Turan, which is already present in [ 8 ] [ 8 ] [8][8][8].
By Gallagher's zero-one law, it is enough to show that meas ( A ) > 0 A ∗ > 0 (A^(**)) > 0\left(\mathcal{A}^{*}\right)>0(A∗)>0. Since q Q A q A ⋃ q ⩾ Q   A q ∗ ↘ A ∗ uuu_(q >= Q)A_(q)^(**)↘A^(**)\bigcup_{q \geqslant Q} \mathcal{A}_{q}^{*} \searrow \mathcal{A}^{*}⋃q⩾QAq∗↘A∗, we may equivalently prove that there is some constant c > 0 c > 0 c > 0c>0c>0 such that meas ( q Q A q ) c ⋃ q ⩾ Q   A q ∗ ⩾ c (uuu_(q >= Q)A_(q)^(**)) >= c\left(\bigcup_{q \geqslant Q} \mathcal{A}_{q}^{*}\right) \geqslant c(⋃q⩾QAq∗)⩾c for all large Q Q QQQ. In order to limit the potential overlap among the sets A q A q ∗ A_(q)^(**)\mathcal{A}_{q}^{*}Aq∗, we only consider an appropriate subset of them. Since meas ( A q ) = 2 φ ( q ) Δ q [ 0 , 1 ] A q ∗ = 2 φ ( q ) Δ q ∈ [ 0 , 1 ] (A_(q)^(**))=2varphi(q)Delta_(q)in[0,1]\left(\mathscr{A}_{q}^{*}\right)=2 \varphi(q) \Delta_{q} \in[0,1](Aq∗)=2φ(q)Δq∈[0,1] for all q q qqq, and since q Q φ ( q ) Δ q = ∑ q ⩾ Q   φ ( q ) Δ q = ∞ sum_(q >= Q)varphi(q)Delta_(q)=oo\sum_{q \geqslant Q} \varphi(q) \Delta_{q}=\infty∑q⩾Qφ(q)Δq=∞, there must exist some R Q R ⩾ Q R >= QR \geqslant QR⩾Q such that
(3.1) 1 q [ Q , R ] meas ( A q ) 2 (3.1) 1 ⩽ ∑ q ∈ [ Q , R ]   meas ⁡ A q ∗ ⩽ 2 {:(3.1)1 <= sum_(q in[Q,R])meas(A_(q)^(**)) <= 2:}\begin{equation*} 1 \leqslant \sum_{q \in[Q, R]} \operatorname{meas}\left(\mathcal{A}_{q}^{*}\right) \leqslant 2 \tag{3.1} \end{equation*}(3.1)1⩽∑q∈[Q,R]meas⁡(Aq∗)⩽2
We will only use the events A q A q ∗ A_(q)^(**)\mathscr{A}_{q}^{*}Aq∗ with q [ Q , R ] q ∈ [ Q , R ] q in[Q,R]q \in[Q, R]q∈[Q,R]. We trivially have the union bound
meas ( q [ Q , R ] A q ) q [ Q , R ] meas ( A q ) 2 meas ⁡ ⋃ q ∈ [ Q , R ]   A q ∗ ⩽ ∑ q ∈ [ Q , R ]   meas ⁡ A q ∗ ⩽ 2 meas(uuu_(q in[Q,R])A_(q)^(**)) <= sum_(q in[Q,R])meas(A_(q)^(**)) <= 2\operatorname{meas}\left(\bigcup_{q \in[Q, R]} \mathcal{A}_{q}^{*}\right) \leqslant \sum_{q \in[Q, R]} \operatorname{meas}\left(\mathcal{A}_{q}^{*}\right) \leqslant 2meas⁡(⋃q∈[Q,R]Aq∗)⩽∑q∈[Q,R]meas⁡(Aq∗)⩽2
If we can show that the sets A q A q ∗ A_(q)^(**)\mathscr{A}_{q}^{*}Aq∗ with q [ Q , R ] q ∈ [ Q , R ] q in[Q,R]q \in[Q, R]q∈[Q,R] do not overlap too much, so that
(3.2) meas ( q [ Q , R ] A q ) c q [ Q , R ] meas ( A q ) c (3.2) meas ⁡ ⋃ q ∈ [ Q , R ]   A q ∗ ⩾ c ∑ q ∈ [ Q , R ]   meas ⁡ A q ∗ ⩾ c {:(3.2)meas(uuu_(q in[Q,R])A_(q)^(**)) >= csum_(q in[Q,R])meas(A_(q)^(**)) >= c:}\begin{equation*} \operatorname{meas}\left(\bigcup_{q \in[Q, R]} \mathcal{A}_{q}^{*}\right) \geqslant c \sum_{q \in[Q, R]} \operatorname{meas}\left(\mathcal{A}_{q}^{*}\right) \geqslant c \tag{3.2} \end{equation*}(3.2)meas⁡(⋃q∈[Q,R]Aq∗)⩾c∑q∈[Q,R]meas⁡(Aq∗)⩾c
we will be able to deduce that meas ( q [ Q , R ] A q ) c meas ⁡ ⋃ q ∈ [ Q , R ]   A q ∗ ⩾ c meas(uuu_(q in[Q,R])A_(q)^(**)) >= c\operatorname{meas}\left(\bigcup_{q \in[Q, R]} \mathcal{A}_{q}^{*}\right) \geqslant cmeas⁡(⋃q∈[Q,R]Aq∗)⩾c and a fortiori that meas ( q Q A q ) c ⋃ q ⩾ Q   A q ∗ ⩾ c (uuu_(q >= Q)A_(q)^(**)) >= c\left(\bigcup_{q \geqslant Q} \mathscr{A}_{q}^{*}\right) \geqslant c(⋃q⩾QAq∗)⩾c. As the following lemma shows, (3.2) is true under (3.1) as long as we can control the correlations of the events A q A q ∗ A_(q)^(**)\mathcal{A}_{q}^{*}Aq∗ on average.
Lemma 3.1. Let E 1 , , E k E 1 , … , E k E_(1),dots,E_(k)E_{1}, \ldots, E_{k}E1,…,Ek be events in the probability space ( Ω , F , P ) ( Ω , F , P ) (Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P})(Ω,F,P). We then have that
P ( j = 1 k E j ) ( j = 1 k P ( E j ) ) 2 i , j = 1 k P ( E i E j ) P ⋃ j = 1 k   E j ⩾ ∑ j = 1 k   P E j 2 ∑ i , j = 1 k   P E i ∩ E j P(uuu_(j=1)^(k)E_(j)) >= ((sum_(j=1)^(k)P(E_(j)))^(2))/(sum_(i,j=1)^(k)P(E_(i)nnE_(j)))\mathbb{P}\left(\bigcup_{j=1}^{k} E_{j}\right) \geqslant \frac{\left(\sum_{j=1}^{k} \mathbb{P}\left(E_{j}\right)\right)^{2}}{\sum_{i, j=1}^{k} \mathbb{P}\left(E_{i} \cap E_{j}\right)}P(⋃j=1kEj)⩾(∑j=1kP(Ej))2∑i,j=1kP(Ei∩Ej)
Proof. Let N = j 1 E j N = ∑ j   1 E j N=sum_(j)1_(E_(j))N=\sum_{j} 1_{E_{j}}N=∑j1Ej. We then have E [ N ] = j P ( E j ) E [ N ] = ∑ j   P E j E[N]=sum_(j)P(E_(j))\mathbb{E}[N]=\sum_{j} \mathbb{P}\left(E_{j}\right)E[N]=∑jP(Ej). On the other hand, the CauchySchwarz inequality implies that
E [ N ] 2 = E [ 1 N > 0 N ] 2 P ( supp ( N ) ) E [ N 2 ] E [ N ] 2 = E 1 N > 0 ⋅ N 2 ⩽ P ( supp ⁡ ( N ) ) ⋅ E N 2 E[N]^(2)=E[1_(N > 0)*N]^(2) <= P(supp(N))*E[N^(2)]\mathbb{E}[N]^{2}=\mathbb{E}\left[1_{N>0} \cdot N\right]^{2} \leqslant \mathbb{P}(\operatorname{supp}(N)) \cdot \mathbb{E}\left[N^{2}\right]E[N]2=E[1N>0⋅N]2⩽P(supp⁡(N))⋅E[N2]
Since supp ( N ) = j E j supp ⁡ ( N ) = ⋃ j   E j supp(N)=uuu_(j)E_(j)\operatorname{supp}(N)=\bigcup_{j} E_{j}supp⁡(N)=⋃jEj and N 2 = i , j 1 E i E j N 2 = ∑ i , j   1 E i ∩ E j N^(2)=sum_(i,j)1_(E_(i)nnE_(j))N^{2}=\sum_{i, j} 1_{E_{i} \cap E_{j}}N2=∑i,j1Ei∩Ej, the lemma follows.
The following proposition summarizes the discussion of this section.
Proposition 3.2. Let Δ 1 , Δ 2 , 0 Δ 1 , Δ 2 , ⋯ ⩾ 0 Delta_(1),Delta_(2),cdots >= 0\Delta_{1}, \Delta_{2}, \cdots \geqslant 0Δ1,Δ2,⋯⩾0, and let A q A q ∗ A_(q)^(**)\mathscr{A}_{q}^{*}Aq∗ be as in (2.5).
(a) If C > 0 C > 0 C > 0C>0C>0 and R Q 1 R ⩾ Q ⩾ 1 R >= Q >= 1R \geqslant Q \geqslant 1R⩾Q⩾1 are such that
(3.3) 1 q [ Q , R ] meas ( A q ) 2 and Q q < r R meas ( A q A r ) C (3.3) 1 ⩽ ∑ q ∈ [ Q , R ]   meas ⁡ A q ∗ ⩽ 2  and  ∑ Q ⩽ q < r ⩽ R   meas ⁡ A q ∗ ∩ A r ∗ ⩽ C {:(3.3)1 <= sum_(q in[Q,R])meas(A_(q)^(**)) <= 2quad" and "quadsum_(Q <= q < r <= R)meas(A_(q)^(**)nnA_(r)^(**)) <= C:}\begin{equation*} 1 \leqslant \sum_{q \in[Q, R]} \operatorname{meas}\left(\mathcal{A}_{q}^{*}\right) \leqslant 2 \quad \text { and } \quad \sum_{Q \leqslant q<r \leqslant R} \operatorname{meas}\left(\mathcal{A}_{q}^{*} \cap \mathcal{A}_{r}^{*}\right) \leqslant C \tag{3.3} \end{equation*}(3.3)1⩽∑q∈[Q,R]meas⁡(Aq∗)⩽2 and ∑Q⩽q<r⩽Rmeas⁡(Aq∗∩Ar∗)⩽C
then meas ( q [ Q , R ] A q ) 1 / ( 2 + 2 C ) meas ⁡ ⋃ q ∈ [ Q , R ]   A q ∗ ⩾ 1 / ( 2 + 2 C ) meas(uuu_(q in[Q,R])A_(q)^(**)) >= 1//(2+2C)\operatorname{meas}\left(\bigcup_{q \in[Q, R]} \mathcal{A}_{q}^{*}\right) \geqslant 1 /(2+2 C)meas⁡(⋃q∈[Q,R]Aq∗)⩾1/(2+2C).
(b) If there are infinitely many disjoint intervals [ Q , R ] [ Q , R ] [Q,R][Q, R][Q,R] satisfying (3.3) with the same constant C > 0 C > 0 C > 0C>0C>0, then meas( lim sup q A q ) = 1 lim sup q → ∞   A q ∗ = 1 {: lims u p_(q rarr oo)A_(q)^(**))=1\left.\lim \sup _{q \rightarrow \infty} \mathscr{A}_{q}^{*}\right)=1limsupq→∞Aq∗)=1.

3.2. A bound on the pairwise correlations

As per Proposition 3.2, we need to control the correlations of the events A q A q ∗ A_(q)^(**)\mathcal{A}_{q}^{*}Aq∗. To this end, we have a lemma of Pollington-Vaughan [22] (see also [10, 27]).
Lemma 3.3. Let q , r q , r q,rq, rq,r be two distinct integers 2 ⩾ 2 >= 2\geqslant 2⩾2, let Δ q , Δ r 0 Δ q , Δ r ⩾ 0 Delta_(q),Delta_(r) >= 0\Delta_{q}, \Delta_{r} \geqslant 0Δq,Δr⩾0, let A q , A r A q ∗ , A r ∗ A_(q)^(**),A_(r)^(**)\mathcal{A}_{q}^{*}, \mathcal{A}_{r}^{*}Aq∗,Ar∗ be as in (2.5), and let M ( q , r ) = 2 max { Δ q , Δ r } lcm [ q , r ] M ( q , r ) = 2 max Δ q , Δ r lcm ⁡ [ q , r ] M(q,r)=2max{Delta_(q),Delta_(r)}lcm[q,r]M(q, r)=2 \max \left\{\Delta_{q}, \Delta_{r}\right\} \operatorname{lcm}[q, r]M(q,r)=2max{Δq,Δr}lcm⁡[q,r]. If M ( q , r ) 1 M ( q , r ) ⩽ 1 M(q,r) <= 1M(q, r) \leqslant 1M(q,r)⩽1, then A q A r = A q ∗ ∩ A r ∗ = ∅ A_(q)^(**)nnA_(r)^(**)=O/\mathscr{A}_{q}^{*} \cap \mathscr{A}_{r}^{*}=\emptysetAq∗∩Ar∗=∅. Otherwise,
meas ( A q A r ) φ ( q ) Δ q φ ( r ) Δ r exp ( p q r / gcd ( q , r ) p > M ( q , r ) 1 p ) meas ⁡ A q ∗ ∩ A r ∗ ≪ φ ( q ) Δ q â‹… φ ( r ) Δ r â‹… exp ⁡ ∑ p ∣ q r / gcd ⁡ ( q , r ) p > M ( q , r )   1 p meas(A_(q)^(**)nnA_(r)^(**))≪varphi(q)Delta_(q)*varphi(r)Delta_(r)*exp(sum_({:[p∣qr//gcd(q","r)],[p > M(q","r)]:})(1)/(p))\operatorname{meas}\left(\mathscr{A}_{q}^{*} \cap \mathscr{A}_{r}^{*}\right) \ll \varphi(q) \Delta_{q} \cdot \varphi(r) \Delta_{r} \cdot \exp \left(\sum_{\substack{p \mid q r / \operatorname{gcd}(q, r) \\ p>M(q, r)}} \frac{1}{p}\right)meas⁡(Aq∗∩Ar∗)≪φ(q)Δq⋅φ(r)Δrâ‹…exp⁡(∑p∣qr/gcd⁡(q,r)p>M(q,r)1p)
Proof. Let Δ = max { Δ q , Δ r } , δ = min { Δ q , Δ r } Δ = max Δ q , Δ r , δ = min Δ q , Δ r Delta=max{Delta_(q),Delta_(r)},delta=min{Delta_(q),Delta_(r)}\Delta=\max \left\{\Delta_{q}, \Delta_{r}\right\}, \delta=\min \left\{\Delta_{q}, \Delta_{r}\right\}Δ=max{Δq,Δr},δ=min{Δq,Δr} and M = M ( q , r ) M = M ( q , r ) M=M(q,r)M=M(q, r)M=M(q,r). The intervals I a = ( a q Δ q , a q + Δ q ) I a = a q − Δ q , a q + Δ q I_(a)=((a)/(q)-Delta_(q),(a)/(q)+Delta_(q))I_{a}=\left(\frac{a}{q}-\Delta_{q}, \frac{a}{q}+\Delta_{q}\right)Ia=(aq−Δq,aq+Δq) and J b = ( b r Δ r , b r + Δ r ) J b = b r − Δ r , b r + Δ r J_(b)=((b)/(r)-Delta_(r),(b)/(r)+Delta_(r))J_{b}=\left(\frac{b}{r}-\Delta_{r}, \frac{b}{r}+\Delta_{r}\right)Jb=(br−Δr,br+Δr) intersect only if 2 Δ > | a q b r | 2 Δ > a q − b r 2Delta > |(a)/(q)-(b)/(r)|2 \Delta>\left|\frac{a}{q}-\frac{b}{r}\right|2Δ>|aq−br|. Since the right-hand side is 1 / lcm [ q , r ] ⩾ 1 / lcm ⁡ [ q , r ] >= 1//lcm[q,r]\geqslant 1 / \operatorname{lcm}[q, r]⩾1/lcm⁡[q,r] when gcd ( a , q ) = gcd ( b , r ) = 1 gcd ⁡ ( a , q ) = gcd ⁡ ( b , r ) = 1 gcd(a,q)=gcd(b,r)=1\operatorname{gcd}(a, q)=\operatorname{gcd}(b, r)=1gcd⁡(a,q)=gcd⁡(b,r)=1, we infer that A q A r = A q ∗ ∩ A r ∗ = ∅ A_(q)^(**)nnA_(r)^(**)=O/\mathcal{A}_{q}^{*} \cap \mathcal{A}_{r}^{*}=\emptysetAq∗∩Ar∗=∅ if M 1 M ⩽ 1 M <= 1M \leqslant 1M⩽1.
Now, assume that M > 1 M > 1 M > 1M>1M>1. Since meas ( I a J b ) 2 δ I a ∩ J b ⩽ 2 δ (I_(a)nnJ_(b)) <= 2delta\left(I_{a} \cap J_{b}\right) \leqslant 2 \delta(Ia∩Jb)⩽2δ for all a , b a , b a,ba, ba,b, we have
meas ( A q A r ) 2 δ # { 1 a q , gcd ( a , q ) = 1 1 b r , gcd ( b , r ) = 1 : | a q b r | < 2 Δ } meas ⁡ A q ∗ ∩ A r ∗ ⩽ 2 δ â‹… # 1 ⩽ a ⩽ q , gcd ⁡ ( a , q ) = 1 1 ⩽ b ⩽ r , gcd ⁡ ( b , r ) = 1 : a q − b r < 2 Δ meas(A_(q)^(**)nnA_(r)^(**)) <= 2delta*#{[1 <= a <= q",",gcd(a","q)=1],[1 <= b <= r",",gcd(b","r)=1]:|(a)/(q)-(b)/(r)| < 2Delta}\operatorname{meas}\left(\mathscr{A}_{q}^{*} \cap \mathscr{A}_{r}^{*}\right) \leqslant 2 \delta \cdot \#\left\{\begin{array}{ll} 1 \leqslant a \leqslant q, & \operatorname{gcd}(a, q)=1 \\ 1 \leqslant b \leqslant r, & \operatorname{gcd}(b, r)=1 \end{array}:\left|\frac{a}{q}-\frac{b}{r}\right|<2 \Delta\right\}meas⁡(Aq∗∩Ar∗)⩽2δ⋅#{1⩽a⩽q,gcd⁡(a,q)=11⩽b⩽r,gcd⁡(b,r)=1:|aq−br|<2Δ}
Let a / q b / r = m / lcm [ q , r ] a / q − b / r = m / lcm ⁡ [ q , r ] a//q-b//r=m//lcm[q,r]a / q-b / r=m / \operatorname{lcm}[q, r]a/q−b/r=m/lcm⁡[q,r]. Then 1 | m | M 1 ⩽ | m | ⩽ M 1 <= |m| <= M1 \leqslant|m| \leqslant M1⩽|m|⩽M and gcd ( m , q 1 r 1 ) = 1 gcd ⁡ m , q 1 r 1 = 1 gcd(m,q_(1)r_(1))=1\operatorname{gcd}\left(m, q_{1} r_{1}\right)=1gcd⁡(m,q1r1)=1, where q 1 = q / d q 1 = q / d q_(1)=q//dq_{1}=q / dq1=q/d and r 1 = r / d r 1 = r / d r_(1)=r//dr_{1}=r / dr1=r/d with d = gcd ( q , r ) d = gcd ⁡ ( q , r ) d=gcd(q,r)d=\operatorname{gcd}(q, r)